Weak null singularities in general relativity

Abstract

We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk and Rodnianski, and conjecturally they are present in the interior of generic black holes arising from gravitational collapse.

1. Introduction

In this paper we study the existence and stability of weak null singularities in general relativity without symmetry assumptions. More precisely, a weak null singularity is a singular null boundary of a spacetime $(\mathcal{M}, g)$ solving the Einstein equations

$$\mathrm{Ric}_{\mu \nu }-\frac{1}{2} R g_{\mu \nu } = T_{\mu \nu }$$

such that the Christoffel symbols blow up and are not square integrable while the metric is continuous up to the boundary. This can be interpreted as a terminal singularity of the spacetime as it cannot be made sense of as a weak solution⁠¹ to the Einstein equations along the singular boundary. While the singularity is sufficiently strong to be terminal, it is at the same time sufficiently weak such that the metric in an appropriate coordinate system is continuous up to the boundary.

¹

One can define a weak solution to the Einstein equations by requiring $\int _{\mathcal{M}} \mathrm{Ric}(X,Y) -\frac{1}{2} R g(X,Y)-T(X,Y) d\mathrm{Vol\,}=0$ in the weak sense for all compactly supported smooth vector fields $X$ and $Y$. After integration by parts, the minimal regularity required for the spacetime for this to be defined is that the Christoffel symbols are square integrable; see the discussion in 5, p.13.

✖

The study of weak null singularities began with the attempts to understand the (in)stability of the Cauchy horizon in the black hole interior of Reissner–Nordström spacetimes. Reissner–Nordström spacetimes are the unique two-parameter family of asymptotically flat (with two ends), spherically symmetric, static solutions to the Einstein–Maxwell equations. Their Penrose diagrams⁠² are given by Figure 1. As seen in the Penrose diagram, the Reissner–Nordström solution possesses a smooth Cauchy horizon $\mathcal{C}\mathcal{H}^+$ in the interior of the black hole such that the spacetime can be extended nonuniquely as a smooth solution to the Einstein–Maxwell system. This feature is also shared⁠³ by the Kerr family of solutions to the vacuum Einstein equations, which can also be depicted by a Penrose diagram given by Figure 1. According to the strong cosmic censorship conjecture (see section 1.1 below), the Reissner–Nordström and Kerr spacetimes are expected to be nongeneric and the smooth Cauchy horizons are expected to be unstable.

²

for $0<{|Q|}<M$

✖

³

for $0<|a|<M$

✖

In a seminal work Dafermos 78 showed that for a spacetime solution to the spherically symmetric Einstein–Maxwell–real scalar field system, if an appropriate upper and lower bound for the scalar field is assumed on the event horizon, then in a neighborhood of timelike infinity, the black hole terminates in a weak null singularity. The necessary upper bound was shown to hold for nonextremal black hole spacetimes arising from asymptotically flat initial data by Dafermos and Rodnianski 10. In particular this implies that near timelike infinity, the terminal boundary of the Cauchy development does not contain a spacelike portion.

In a more recent work 9, Dafermos showed that if, in addition to assuming the two black hole exterior regions settle to Reissner–Nordström with appropriate rates, the initial data are moreover globally close to that of Reissner–Nordström, then the maximal Cauchy development of the data possesses the same Penrose diagram as Reissner–Nordström. In particular the spacetime terminates in a global bifurcate weak null singularity and the singular boundary does not contain any spacelike portion.

The works 789 were in part motivated by the physics literature on the instability of Cauchy horizons, weak null singularities and the strong cosmic censorship conjecture. It will be discussed below in section 1.1.

While the works of Dafermos 789 are restricted to the class of spherically symmetric spacetimes, they nonetheless suggest the genericity of weak null singularities in the black hole interior, at least “in a neighborhood of timelike infinity”. In particular they motivate the following conjecture for the vacuum Einstein equations,

$$\begin{equation} \mathrm{Ric}_{\mu \nu }=0. {\tag{1}} \end{equation}$$

Conjecture 1.

(1)

Consider the characteristic initial value problem with smooth characteristic initial data on a pair of null hypersurfaces $H_0$ and $\underline{H}_0$ intersecting on a $2$-sphere. Suppose that $H_0$ is an affine complete null hypersurface on which the data approach that of the event horizon of a Kerr solution (with $0<|a|<M)$ at a sufficiently fast polynomial rate.⁠⁴ Then the development $(\mathcal{M}, g)$ of the initial data possesses a null boundary “emanating from timelike infinity $i_+$” through which the spacetime is extendible with a continuous metric (see shaded region in Figure 2). Moreover, given an appropriate “lower bound” on the $H_0$, this piece of null boundary is generically a weak null singularity with nonsquare-integrable Christoffel symbols.

⁴

In particular this applies if an asymptotically flat spacetime has an exterior region which approaches a subextremal Kerr solution at a sufficiently fast polynomial rate. This also holds in the case where the Cauchy hypersurface has only one asymptotically flat end. In that case, numerical work in spherical symmetry 13 suggests that the singular boundary may also contain a nonempty spacelike portion, in addition to the null portion.

✖

(2)

(Ori, see discussion in 9) If the data for $(\mathcal{M}, g)$ on a complete two-ended asymptotically flat Cauchy hypersurface are globally a small perturbation of two-ended Kerr initial data (with $0<{|a|}<M$), then the maximal Cauchy development possesses a global bifurcate future null boundary $\partial \mathcal{M}$. Moreover, for generic such perturbations of Kerr, $\partial \mathcal{M}$ is a global bifurcate weak null singularity which intersects every futurely causally incomplete geodesic.

If Conjecture 1 is true, then in particular there exist local stable weak null singularities for the vacuum Einstein equations without symmetry assumptions. We show in this paper that there is in fact a large class of such singularities, parameterized by singular initial data. More specifically, we solve a characteristic initial value problem with singular initial data and construct a class of stable bifurcate weak null singularities.

To motivate the strength of the singularity considered in this paper, we first recall the strength of the spherically symmetric weak null singularities in a neighborhood of Reissner–Nordström studied in 8. The instability of the Reissner–Nordström Cauchy horizon is in fact already suggested by a linear analysis (see 42023). For a spherically symmetric solution to the linear wave equation which has a polynomially decaying (in the Eddington–Finkelstein coordinates) tail⁠⁵ along the event horizon, there is a singularity in a ($C^0$)-regular coordinate system near the Cauchy horizon of the strength⁠⁶

⁵

with upper and lower bounds

✖

⁶

This statement regarding the linear wave equation can be inferred using the methods in 7 for the nonlinear coupled Einstein–Maxwell–scalar field system.

✖

$$\begin{eqnarray} |\partial _{\underline{u}} \phi |\sim (\underline{u}_*-\underline{u})^{-1}{\log ^{-p}\left(\frac{1}{\underline{u}_*-\underline{u}}\right)}, {\tag{2}} \end{eqnarray}$$

for some $p>1$ as $\underline{u}\to \underline{u}_*$. In particular along an outgoing null curve, $\partial _{\underline{u}}\phi$ is integrable but not $L^q$-integrable for any $q>1$. In the spacetimes constructed by Dafermos 78, it was shown moreover that even in the nonlinear setting, $\partial _{\underline{u}}\phi$ is also singular but remains integrable. A more precise analysis will show that in fact the spherically symmetric scalar field in the nonlinear setting of 8 also blows up at a rate given by 2.

Returning to the problem of constructing stable weak null singularities in vacuum, our construction is based on solving a characteristic initial value problem with singular data. We will in fact construct spacetimes not only with one weak null singularity, but instead they will contain two weak null singularities terminating at a bifurcate sphere. More precisely, the data on the initial characteristic hypersurface $H_0$ (resp. $\underline{H}_0$) is determined by the traceless part of the null second fundamental form $\hat{\chi }$ (resp. $\hat{\underline{\chi }}$). We consider singular initial data satisfying in particular

$$|\hat{\chi }|\sim (\underline{u}_*-\underline{u})^{-1}{\log ^{-p}\left(\frac{1}{\underline{u}_*-\underline{u}}\right)},\quad \text{for some }p>1,$$

and

$$|\hat{\underline{\chi }}|\sim (u_*-u)^{-1}{\log ^{-p}\left(\frac{1}{u_*-u}\right)},\quad \text{for some }p>1.$$

This singularity is consistent with the strength of the weak null singularities in 2.

The following is a first version of the main result of this paper (see Figure 3). We refer the readers to the statement of Theorems 2, 3 and 4 for a more precise formulation of the theorem.

Theorem 1 (Main theorem, first version).

For a class of singular characteristic initial data without any symmetry assumptions for the vacuum Einstein equations

$$\mathrm{Ric}_{\mu \nu }=0$$

with the singular profile as above (see precise requirements on the data in section 1.3) and for $\epsilon$ sufficiently small and $u_*$, $\underline{u}_*\leq \epsilon$, there exists a unique smooth spacetime $(\mathcal{M}, g)$ endowed with a double null foliation $(u,\underline{u})$ in $0\leq u<u_*$, $0\leq \underline{u}< \underline{u}_*$, which satisfies the vacuum Einstein equations with the given data. Associated to $(\mathcal{M}, g)$, there exists a coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$ such that the metric extends continuously to the boundary but the Christoffel symbols are not in $L^2$.

Remark 1.

This class of stable local weak null singularities that we construct in particular provides the first construction of weak null singularities of such strength for the vacuum Einstein equations.⁠⁷

⁷

We recall Birkhoff’s theorem which states that the only spherically symmetric vacuum spacetimes are the Minkowski and Schwarzschild solutions. Thus to construct stable examples of weak null singularities in vacuum, one necessarily works outside the class of spherically symmetric spacetimes.

✖

Theorem 1 allows singularities on both initial null hypersurface and is valid in the region where $u_*$ and $\underline{u}_*$ are sufficiently small. In the context of the interior of black holes, this corresponds to the darker shaded region in Figure 4. The existence theorem clearly implies an existence result when the data are only singular on one of the initial null hypersurfaces. In that context, we can in fact combine the methods in this paper with that in 17 to show that the domain of existence can be extended so that only one of the characteristic length scales is required to be small. More precisely, we allow that data on $H_0$ such that

$$|\hat{\chi }| \sim (\underline{u}_*-\underline{u})^{-1}{\log ^{-p}\left(\frac{1}{\underline{u}_*-\underline{u}}\right)},\quad \text{for some }p>1,$$

on $0\leq \underline{u}<\underline{u}_*\leq C$ and the data on $\underline{H}_0$ are smooth on $0\leq u\leq u_*\leq \epsilon$. Then for $\epsilon$ sufficiently small, the spacetime $(\mathcal{M}, g)$ remains smooth in $0\leq u<u_*$, $0\leq \underline{u}< \underline{u}_*$ (see for example the lightly shaded region in Figure 4). We will omit the details of the proof of this result.

Theorem 1, which proves the existence and stability of the conjecturally generic weak null singularities, can be viewed as a first step toward Conjecture 1. A next step is an analogue of 8 for the vacuum Einstein equations without symmetry assumptions, i.e., to solve the characteristic initial value problem inside the black hole with data prescribed on the event horizon that is approaching Kerr at appropriate rates. This requires an understanding of the formation of weak null singularities from smooth data on the event horizon (see part (1) of Conjecture 1). A full resolution of Conjecture 1, part (2), however, requires in addition an understanding of the decay rates of gravitational radiation along the event horizon for generic perturbations of Kerr spacetime. This latter problem is intimately tied to the problem of the nonlinear stability of Kerr spacetimes, which continues to be one of the most important and challenging open problems in mathematical general relativity. Nevertheless, significant progress has been made for the corresponding linear problem in the past decade. We refer the readers to the survey of Dafermos and Rodnianski 11 for more about this linear problem.

The approach for the main theorem applies equally well to the Einstein–Maxwell–scalar field system without symmetry assumptions.⁠⁸ Thus, we show that the weak null singularity of Dafermos 8, which arises from appropriately decaying data on the event horizon, is stable against nonspherically symmetric perturbations on the hypersurface $\Sigma$ sufficiently far within the black hole region (see Figure 5).

⁸

This can be easily seen by decomposing the Maxwell field and the gradient of the scalar field in terms of the null frame below. The components in this decomposition obey equations that can be put in the same schematic form as in section 2.4. Therefore, the Maxwell field and the scalar field and their derivatives satisfy estimates similar to those for the Ricci coefficients and curvature components.

✖

1.1. Weak null singularities and strong cosmic censorship conjecture

The study of weak null singularities can be viewed in the larger context of Penrose’s celebrated strong cosmic censorship conjecture in general relativity. The conjecture states that for generic asymptotically flat initial data for “reasonable” Einstein-matter systems, the maximal Cauchy development is future inextendible as a suitably regular Lorentzian manifold. This would guarantee general relativity to be a deterministic theory.

As pointed out above, the Kerr and Reissner–Nordström families of solutions (of the Einstein vacuum and Einstein–Maxwell equations, respectively) have maximal Cauchy developments that are extendible as larger smooth spacetimes unless the angular momentum or the charge vanishes. This is connected with the existence of a smooth Cauchy horizon in the black hole interior such that the spacetime can be extended beyond as a smooth solution. According to the strong cosmic censorship conjecture, this is expected to be nongeneric.

On the other hand, the situation for the Schwarzschild spacetime is more preferable from the point of view of the deterministic nature of the theory. The maximal development of the Schwarzschild spacetime terminates with a spacelike singularity at which the Hawking mass and the curvature scalar invariants blow up. In particular the spacetime cannot be extended in $C^2$.

The early motivation for the strong cosmic censorship conjecture, besides the desirability of a deterministic theory, is a linear heuristic argument by Penrose 23 suggesting that the Reissner–Nordström Cauchy horizon is unstable. This was also confirmed by the numerical work by Simpson and Penrose 27. It is thus conjectured that a small global perturbation would lead to a singularity in the interior of the black hole in such a way that the maximal Cauchy development is future inextendible.

However, the nature of the singular boundary in the interior of black holes was not well understood⁠⁹ until the first study of weak null singularity carried out by Hiscock 12. In an attempt to understand the instability of the Reissner–Nordström Cauchy horizon, he considered the Vaidya model allowing for a self-gravitating ingoing null dust. In this model, an explicit solution can be found, and he showed that various components of the Christoffel symbols blow up. This, however, was called a whimper singularity as the Hawking mass and the curvature scalar invariants remain bounded.

⁹

In particular it was believed that a perturbation of the Reissner–Nordström Cauchy horizon would lead to a Schwarzschild type singularity.

✖

In subsequent works, Poisson and Israel 2526 added an outgoing null dust to the model considered by Hiscock. While explicit solutions were not available, they were able to deduce that the second outgoing null dust would cause the Hawking mass to blow up at the null singularity. It was then thought of as a stronger singularity than that of Hiscock.

However, from the point of view of partial differential equations, it is more natural to view this singularity at the level of the nonsquare-integrability of the Christoffel symbols, which is exactly the threshold such that the spacetime cannot be defined as a weak solution to the Einstein equations. From this perspective, the singularity of Poisson and Israel is as strong as that of Hiscock, and both singularities can be viewed as terminal boundaries for the spacetimes in question.

While the Christoffel symbols blow up at the Cauchy horizon, one can also think that the Cauchy horizon is “stable” in the sense that no singularity arises before the “original Cauchy horizon”. In particular there is no spacelike portion of the singular boundary in a neighborhood of timelike infinity. Thus, this is contrary to the case of the Schwarzschild spacetime. This weak null singularity picture has been further explored and justified in many numerical works (see 123).

As we described before, the aforementioned picture of the interior of black holes was finally established by Dafermos in the context of the spherically symmetric Einstein–Maxwell–scalar field system 7. This is the main motivation for our present work in which we initiate the study of weak null singularities of similar strength in vacuum without any symmetry assumptions.

Finally, we note that a class of analytic spacetimes with slightly weaker singularities have been previously constructed in 22. While this class of spacetime is more restrictive, as discussed in 22, it nonetheless admits the full “functional degrees of freedom” of the Einstein equations.

1.2. Comparison with impulsive gravitational waves

As pointed out by Dafermos 9, the weak null singularities that we consider in this paper share many similarities with impulsive gravitational waves. The latter are vacuum spacetimes admitting null hypersurfaces which support delta function singularities in the Riemann curvature tensor. Explicit examples were first constructed by Penrose 24, Khan and Penrose 14, and Szekeres 28. In these spacetimes, while the Christoffel symbols are not continuous, they remain bounded. Therefore, in contrast with the weak null singularities that we consider here, these impulsive gravitational waves are not terminal singularities. In fact, the solution to the vacuum Einstein equation extends beyond the singularity and is smooth except across the singular hypersurface. Nevertheless, both scenarios represent singularities propagating along null hypersurfaces and from a mathematical point of view, the proofs of the existence theory for these singularities share many common features.

In recent joint works with Rodnianski 1819, we initiated the rigorous mathematical study for general impulsive gravitational waves without symmetry assumptions. We constructed the impulsive gravitational waves via solving the characteristic initial problem such that the initial data admit curvature delta singularities supported on an embedded 2-sphere. One of the new ideas in the proof is the use of renormalized energy estimates for the curvature components; i.e., instead of controlling the spacetime curvature components in $L^2$, we subtract off an $L^\infty$ correction from some curvature components. This allowed us to derive a closed system of $L^2$ estimates which is completely independent of the singular curvature components.

In 18, when the interaction of impulsive gravitational waves was studied, we also extended the analysis to include a class of spacetimes such that when measured in the worst direction, the Christoffel symbols are merely in $L^2$. We proved an existence and uniqueness theorem for spacetimes with such low regularity and showed that the spacetime solution can be extended beyond the singularities. Notice that this result is in fact sharp: this is because if the Christoffel symbols fail to be square integrable, the spacetime cannot be extended as a weak solution to the Einstein equations (see footnote 1).

By contrast, the spacetimes considered in this paper have Christoffel symbols which are⁠¹⁰ not in $L^2$. Even though the weak null singularities are terminal singularities in the sense that there cannot be an existence theory beyond them, the theory developed in 1819 can be extended to control the spacetime up to the singularity. Moreover, our main theorem, which allows for two weak null singularities terminating at their intersection, can be viewed as an extension of the result in 18 on the interaction of two impulsive gravitational waves. In particular the renormalized energy of 1819 plays an important role in the proof of our main theorem. However, even after renormalization, the renormalized curvature is still singular (i.e., not in $L^2$) and has to be dealt with using an additional weighted estimate.

¹⁰

In fact, we allow initial data to be in $L^p$ only for $p=1$, but not for any $p>1$.

✖

1.3. Description of the main results

Our setup is the characteristic initial value problem with initial data given on two null hypersurfaces $H_0$ and $\underline{H}_0$ intersecting at a 2-sphere $S_{0,0}$ (see Figure 6). We will follow the general notations in 51516.

We introduce a null frame $\{e_1,e_2,e_3,e_4\}$ adapted to a double null foliation $(u,\underline{u})$ (see section 2.1). Denote the constant $u$ hypersurfaces by $H_u$, the constant $\underline{u}$ hypersurfaces by $\underline{H}_{\underline{u}}$ and their intersections by $S_{u,\underline{u}}=H_u\cap \underline{H}_{\underline{u}}$. Decompose the Riemann curvature tensor with respect to the null frame $\{e_1,e_2,e_3,e_4\}$:

$$\begin{equation*} \begin{split} \alpha _{AB} &=R(e_A, e_4, e_B, e_4),\qquad \underline{\alpha }_{AB}=R(e_A, e_3, e_B, e_3),\\ \beta _A &= \frac{1}{2} R(e_A, e_4, e_3, e_4),\qquad \underline{\beta }_A =\frac{1}{2} R(e_A, e_3, e_3, e_4),\\ \rho &=\frac{1}{4} R(e_4,e_3, e_4, e_3),\qquad \ \ \sigma =\frac{1}{4} \,^*R(e_4,e_3, e_4, e_3){.} \end{split} \end{equation*}$$

We also define the Gauss curvature of the 2-spheres associated to the double null foliation to be $K$. Define also the following Ricci coefficients with respect to the null frame:

$$\begin{equation*} \begin{split} &\chi _{AB}=g(D_A e_4,e_B), \qquad \underline{\chi }_{AB}=g(D_A e_3,e_B),\\ &\eta _A=-\frac{1}{2} g(D_3 e_A,e_4),\qquad \underline{\eta }_A=-\frac{1}{2} g(D_4 e_A,e_3){,}\\ &\omega =-\frac{1}{4} g(D_4 e_3,e_4),\qquad \ \ \,\,\underline{\omega }=-\frac{1}{4} g(D_3 e_4,e_3),\\ &\zeta _A=\frac{1}{2} g(D_A e_4,e_3){.} \end{split} \end{equation*}$$

Let $\hat{\chi }$ (resp. $\hat{\underline{\chi }}$) be the traceless part of $\chi$ (resp. $\underline{\chi }$).

The data on $H_0$ are given on $0\leq \underline{u}< \underline{u}_*$ such that $\chi$ becomes singular as $\underline{u}\to \underline{u}_*$. Similarly, the data on $\underline{H}_0$ is given on $0\leq u < u_*$ such that $\underline{\chi }$ becomes singular as $u \to u_*$.

More precisely, let $f_1:{[0,\underline{u}_*)} \to \mathbb{R}$ be a smooth function such that ${f_1}(x)\geq 0$ is decreasing and

$$\int _0^{{\underline{u}_*}} \frac{1}{f_1(x)^2} dx < \infty$$

(resp. let $f_2:[0,{u_*}) \to \mathbb{R}$ be a smooth function such that ${f_2}(x)\geq 0$ is decreasing and

$$\int _0^{{u_*}} \frac{1}{f_2(x)^2} dx < \infty ).$$

For example, $f_1$ can be taken to be $f_1(x)=({\underline{u}_*}-x)^{\frac{1}{2}}\log ^p (\frac{{1}}{{\underline{u}_*}-x})$ for $p>\frac{1}{2}$.

Our main theorem shows local existence for a class of singular initial data with⁠¹¹

¹¹

We assume also bounds for the angular derivatives that are consistent with this singular profile (see the precise statement in Theorem 2).

✖

$$|\chi (0,\underline{u})|\lesssim f_1(\underline{u})^{-2},\qquad |\underline{\chi }(u,0)|\lesssim f_2(u)^{-2}.$$

We construct a (unique) solution $(\mathcal{M},g)$ to the vacuum Einstein equations in the region $u<u_*$, $\underline{u}<\underline{u}_*$, where $u_*$, $\underline{u}_*\leq \epsilon$, and

$$\begin{eqnarray} \int _0^{\underline{u}_*} f_1(\underline{u})^{-2} d\underline{u},\qquad \int _0^{u_*} f_2(u)^{-2} du\leq \epsilon ^2. {\tag{3}} \end{eqnarray}$$

Here, $(u,\underline{u})$ is a double null foliation for $(\mathcal{M},g)$ and the metric $g$ takes the form

$$g=-2\Omega ^2(du\otimes d\underline{u}+d\underline{u}\otimes du)+\gamma _{AB} (d\theta ^A-b^A du)\otimes (d\theta ^B-b^B du)$$

in the $(u,\underline{u},\theta ^1,\theta ^2)$ coordinate system (to be defined in section 2.2). Define also $\nabla$ to be the induced Levi-Cevita connection on the $2$-spheres of constant $u$ and $\underline{u}$, i.e., $S_{u,\underline{u}}$, and $\nabla _3$, $\nabla _4$ to be the projections of the covariant derivatves $D_3$, $D_4$ to the tangent space of $S_{u,\underline{u}}$. Our main theorem (Theorem 1) can be stated precisely as a combination of Theorems 2, 3 and 4. The first main result is the following theorem, which shows the existence of a spacetime up to the (potentially singular) null boundaries:

Theorem 2.

Consider the characteristic initial value problem for

$$\begin{equation} \mathrm{Ric}_{\mu \nu }=0 {\tag{4}} \end{equation}$$

with data that are smooth on $H_0\cap \{0\leq \underline{u}< \underline{u}_*\}$ and $\underline{H}_0\cap \{0\leq u< u_*\}$ such that the following hold.

•

There exists an atlas such that in each coordinate chart with local coordinates $(\theta ^1,\theta ^2)$, the initial metric $\gamma _0$ on $S_{0,0}$ obeys$$d\leq \det \gamma _0\leq D$$

and$$\sum _{i_1+i_2\leq 6}\left|\left(\frac{\partial }{\partial \theta ^1}\right)^{i_1}\left(\frac{\partial }{\partial \theta ^2}\right)^{i_2} \gamma _{BC}\right|\leq D.$$

•

The metric on $H_0$ and $\underline{H}_0$ satisfies the gauge conditions$$\Omega =1\quad \text{on }H_0\text{ and }\underline{H}_0$$

and$$b^A=0\quad \text{on }\underline{H}_0.$$

•

The Ricci coefficients on the initial hypersurface $H_0$ verify$$\sum _{i\leq 5}\sup _{\underline{u}}\left\|{f_1}^2(\underline{u})\nabla ^i\chi \right\|_{L^2(S_{0,\underline{u}})}\leq D,$$

$$\sum _{i\leq 4}\sup _{\underline{u}}\left\|\nabla ^i\zeta \right\|_{L^2(S_{0,\underline{u}})}\leq D,$$

$$\sum _{i\leq 4}\sup _{\underline{u}}\left\|\nabla ^i\mathrm{tr\,}\underline{\chi }\right\|_{L^2(S_{0,\underline{u}})}\leq D.$$

•

The Ricci coefficients on the initial hypersurface $\underline{H}_0$ verify$$\sum _{i\leq 5}\sup _{u}\left\|{f_2}^2(u)\nabla ^i\underline{\chi }\right\|_{L^2(S_{u,0})}\leq D,$$

$$\sum _{i\leq 4}\sup _{u}\left\|\nabla ^i\zeta \right\|_{L^2(S_{u,0})}\leq D,$$

$$\sum _{i\leq 4}\sup _{u}\left\|\nabla ^i\mathrm{tr\,}\chi \right\|_{L^2(S_{u,0})}\leq D.$$

Then for $\epsilon$ sufficiently small (depending only on $d$ and $D$) and

$$u_*,\, \underline{u}_*\leq \epsilon ,\quad \left\|{f_1}(\underline{u})^{-1}\right\|_{L^2_{\underline{u}}},\,\left\|{f_2}(u)^{-1}\right\|_{L^2_{u}}<\epsilon ,$$

there exists a unique spacetime $(\mathcal{M}, g)$ endowed with a double null foliation $(u,\underline{u})$ in $0\leq u< u_*$ and $0\leq \underline{u}<\underline{u}_*$, which is a solution to the vacuum Einstein equations 4 with the given data. Moreover, the spacetime remains smooth in $0\leq u< u_*$ and $0\leq \underline{u}<\underline{u}_*$.

Remark 2.

In the following, we will only prove a priori estimates for spacetimes arising from these initial data (see Theorem 5). The existence of a spacetime and the propagation of regularity follow from standard arguments. (For an example of this argument in low regularity, see 19, Sections 4 and 5. See also 5, Chapter 16.)

Remark 3.

In order to simplify notations, we will omit the subscripts $1$ and $2$ in the weight functions $f_1$ and $f_2$. They can be inferred from whether $f$ is a function of $u$ or $\underline{u}$.

Remark 4.

In section 4, we will construct a class of characteristic initial data which satisfies the assumptions of Theorem 2.

While the weight $f$ in the spacetime norms allows the spacetime to be singular, the spacetime metric can be extended beyond the singular hypersurfaces $H_{u_*}$ and $\underline{H}_{\underline{u}_*}$ continuously.

Theorem 3.

Under the assumptions of Theorem 2, the spacetime $(\mathcal{M}, g)$ can be extended continuously up to and beyond the singular boundaries $\underline{H}_{\underline{u}_*}:=\{\underline{u}=\underline{u}_*\}$, $H_{u_*}:=\{u=u_*\}$. Moreover, the induced metric and null second fundamental form on the interior of the limiting hypersurfaces $\underline{H}_{\underline{u}_*}$ and $H_{u_*}$ are regular. More precisely, for any coordinate chart $U_i$ on $S_{0,0}$, the metric components $\gamma$, $b$, $\Omega$ satisfy the following estimates in the coordinate chart given by $U_i(u,\underline{u}):=\underline{\Phi }_{\underline{u}}\circ \Phi _u(U_i)$, where $\Phi _u$ and $\underline{\Phi }_{\underline{u}}$ are the diffeomorphisms generated by $\underline{L}$ and $L,$ respectively:⁠¹²

¹²

See definition of $L$ and $\underline{L}$ in section 2.1.

✖

$${\sum _{i_1+i_2\leq 4}} \sup _{0\leq u\leq {u_*}}\left\|\left(\frac{\partial }{\partial \theta ^1}\right)^{i_1}\left(\frac{\partial }{\partial \theta ^2}\right)^{i_2} \left(\gamma ,b,\Omega \right)\right\|_{L^2\left(U_i\left(u,\underline{u}_*\right)\right)} \leq C{.}$$

Moreover, for any fixed $U<u_*$, we have the following bounds for the Ricci coefficients $\hat{\underline{\chi }},\mathrm{tr\,}\underline{\chi },\underline{\omega },\eta ,\underline{\eta }$:

$${\sum _{j\leq 1}\,\sum _{i\leq 3-j}}\sup _{0\leq u\leq U}\left\|\nabla _3^j\nabla ^i\left(\hat{\underline{\chi }},\mathrm{tr\,}\underline{\chi },\underline{\omega },\eta ,\underline{\eta }\right) \right\|_{L^2\left(S_{u,\underline{u}_*}\right)}\leq C_{U}.$$

Similar regularity statements hold on $H_{u_*}$.

Remark 5.

If we assume in addition that the higher angular derivatives of $\chi$ are bounded in $L^1_{\underline{u}}L^\infty (S)$, then the metric and the second fundamental form also inherit higher regularity in the interior of $\underline{H}_{\underline{u}_*}$. In particular if all angular derivatives of $\chi$ are bounded in $L^1_{\underline{u}}L^\infty (S)$, then the metric restricted to $\underline{H}_{\underline{u}_*}\cap \{0\leq u\leq U\}$ is smooth along the directions tangential to $\underline{H}_{\underline{u}_*}$. Similar statements hold on $H_{u_*}$. We will omit the details.

Moreover, we show that if initially the data are indeed singular, then $H_{u_*}$ and $\underline{H}_{\underline{u}_*}$ are terminal singularities of the spacetime in the following sense:

Theorem 4.

If, in addition to the assumptions of Theorem 2, we also have the following for the initial data,

$$\int _0^{\underline{u}_*} \left|\hat{\chi }\left(0,\underline{u}\right)\right|^2 d\underline{u}=\infty ,$$

along Lebesgue-almost every null generator on $H_0$, then the Christoffel symbols in the coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$ do not belong to $L^2$ in a neighborhood of any point on $\underline{H}_{\underline{u}_*}$.

Similarly if the initial data satisfy

$$\int _0^{u_*} \left|\hat{\underline{\chi }}(u,0)\right|^2 du =\infty$$

along Lebesgue-almost every null generator on $\underline{H}_0$, then the Christoffel symbols in the coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$ do not belong to $L^2$ in a neighborhood of any point on $H_{u_*}$.

Remark 6.

Theorem 4 guarantees that if we extend the spacetime metric continuously in the obvious differentiable structure given by the coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$, then the Christoffel symbols are nonsquare-integrable in the extension. However, it is an open problem whether the spacetime admits any continuous extensions with square integrable Christoffel symbols.

1.4. Main ideas of the proof

All the known proofs of regularity for the Einstein equations without symmetry assumptions rely on $L^2$ estimates on the metric and its derivatives or the Riemann curvature tensor and its derivatives. Let us denote schematically by $\Gamma$ a general Ricci coefficient and by $\Psi$ a general curvature component decomposed with respect to a null frame adapted to the double null foliation. In the double null foliation gauge (see, for example, 515), the standard approach to obtain a priori bounds is to couple the $L^2$ estimates for the curvature components

$$\int _H \Psi ^2 +\int _{\underline{H}} \Psi ^2 \leq \text{ Data }+\iint \Gamma \Psi \Psi$$

with the estimates for the Ricci coefficients obtained using the transport equations

$$\nabla _3\Gamma =\Psi +\Gamma \Gamma ,$$

$$\nabla _4\Gamma =\Psi +\Gamma \Gamma .$$ However, in the setting of two weak null singularities, none of the spacetime curvature components $\alpha , \beta , \rho , \sigma , \underline{\beta }, \underline{\alpha }$ are in $L^2$!

Nevertheless, while these curvature components are singular, the nature of their singularity is specific. More precisely, while the spacetime curvature components $\rho$ and $\sigma$ are not in $L^2$, they can be written as a sum of some regular intrinsic curvature components $K$ and $\check{\sigma }$ (see further discussion in section 1.4.1) which belong to $L^2$ and terms which are quadratic in $\Gamma$. We therefore prove $L^2$ estimates for $K$ and $\check{\sigma }$, which we will call the renormalized curvature components (see 1819). Moreover, by considering $(K,\check{\sigma })$ instead of $(\rho ,\sigma )$, we remove all appearances of $\alpha$ and $\underline{\alpha }$ in the estimates and so that we do not have to deal with the singularities of $\alpha$ and $\underline{\alpha }$! It still remains to control the singular curvature components $\beta$ and $\underline{\beta }$. Here, we make use of the fact that $\beta$ and $\underline{\beta }$ are singular in a specific manner toward the singular boundary $\underline{H}_{\underline{u}_*}$ and $H_{u_*},$ respectively. We therefore introduce degenerate $L^2$ norms that incorporate these singularities. We will explain the renormalization and the degenerate estimates in more detail below.

1.4.1. Renormalized energy estimates

As described above, a main ingredient of the proof of the main theorem is the renormalized energy estimates introduced in 1819 in the study of impulsive gravitational waves. This can be seen as follows. For the class of weak null singularities that we consider, while the ${\mathcal{L}} \mkern -9mu /\,_L$ derivative of the spacetime metric blows up, the metric restricted to the $2$-sphere remains regular in the angular directions. Since the Gauss curvature $K$ is intrinsic to the $2$-spheres, it remains bounded. On the other hand, by the Gauss equation,

$$K=-{\rho }+\frac{1}{2} \hat{\chi }\cdot \hat{\underline{\chi }}-\frac{1}{4}\mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi },$$

and the fact that $\mathrm{tr\,}\chi$ and $\hat{\chi }$ blow up at $\underline{u}=\underline{u}_*$, $\rho$ also blows up at $\underline{u}=\underline{u}_*$. In view of this, we estimate the Gauss curvature $K$ instead of the spacetime curvature component $\rho$.

Indeed, we see that the Gauss curvature $K$ satisfies equations such that the right-hand side contains terms that are less singular than the terms in the corresponding equation for $\rho$. More precisely, for the curvature component $\rho$, we have (up to lower-order terms) the Bianchi equation

$$\nabla _4\rho +\frac{3}{2}\mathrm{tr\,}\chi \rho = \mathrm{div\,}\beta -\frac{1}{2} \hat{\underline{\chi }}\cdot \alpha +\cdots ,$$

which contains the nonintegrable curvature component $\alpha$. On the other hand, the Gauss curvature obeys the equation (see 12)

$$\nabla _4 K+\mathrm{tr\,}\chi K= -\mathrm{div\,}\beta +\cdots ,$$

where there are no terms containing $\alpha$ or that are quadratic in $\mathrm{tr\,}\chi$, $\hat{\chi }$ and $\omega$, i.e., every term on the right-hand side of the equation is integrable in the $\underline{u}$ direction.⁠¹³

¹³

The can be compared with the renormalization introduced in 19 and 18, where we estimated $\check{\rho }=\rho -\frac{1}{2} \hat{\chi }\cdot \hat{\underline{\chi }}$ instead of $\rho$. Whereas the renormalization using $\check{\rho }$ allows one to eliminate $\alpha$ in the estimates, it nonetheless introduces a term $\frac{1}{4}\mathrm{tr\,}\underline{\chi }|\hat{\chi }|^2$, which is not integrable in the $\underline{u}$ direction in the setting of the present paper. Instead, by studying the equation for $K$, we see none of these terms which are quadratic in $\mathrm{tr\,}\chi$, $\hat{\chi }$ or $\omega$! This fact can also be derived directly by considering the equations for $\nabla _4 K$ using the intrinsic definition of the Gauss curvature.

✖

In a similar fashion, by considering the renormalized curvature component⁠¹⁴

¹⁴

This is in fact related to the intrinsic curvature of the normal bundle to $S_{u,\underline{u}}$.

✖

$$\check{\sigma }:=\sigma +\frac{1}{2}\hat{\underline{\chi }}\wedge \hat{\chi }$$

instead of $\sigma$, we see that it satisfies an equation such that all the terms on the right-hand side are integrable in the $\underline{u}$ direction.

One consequence of the renormalization is that we have completely removed the appearances of the curvature component $\alpha$ in the equations. In fact, as in 1819, this allows us to derive a set of estimates for the renormalized curvature component without requiring any information on the curvature component $\alpha$.

Moreover, when considering the equations for $\nabla _3 K$ and $\nabla _3\check{\sigma }$ for the renormalized curvature components, one sees that $\underline{\alpha }$ does not appear and all the terms are integrable in the $u$ direction. Therefore, although $\alpha$ or $\underline{\alpha }$ can be very singular near one of the singular boundaries, we do not need to derive any estimates for them!

1.4.2. Degenerate $L^2$ estimates

Since the renormalization above deals with the singularity in the $\rho$ and $\sigma$ components and avoids any information on $\alpha$ and $\underline{\alpha }$, it remains to derive appropriate $L^2$ estimates for $\beta$ and $\underline{\beta }$.

The main observation is that while $\beta$ and $\underline{\beta }$ are both singular and fail to be in $L^2$, their singularities can be captured quantitatively. Consider the curvature component $\beta$. Since the blow-up rate of $\mathrm{tr\,}\chi$ and $\hat{\chi }$ can be bounded above by $f(\underline{u})^{-2}$, in view of the Codazzi equations in 10, $\beta$ is also bounded above by $f(\underline{u})^{-2}$. In particular while $\beta$ is only in $L^1_{\underline{u}}$ but not in $L^p_{\underline{u}}$ for any $p>1$, the assumptions on the initial data allow us to control $f(\underline{u})\beta$ in $L^2_{\underline{u}}$. We will thus incorporate this blowup in the norms and will be able to still use an $L^2$ based estimate.

The energy estimates will be obtained directly from two sets of Bianchi equations instead of using the Bel–Robinson tensor. Notice that since the energy estimates for $K,\check{\sigma }$ are obtained either together with that for $\beta$ or that for $\underline{\beta }$, even though $K$ and $\check{\sigma }$ are regular, their energy estimates degenerate. Therefore, at the highest level of derivatives, we have to be content with the weaker $L^2$ estimates for these curvature components.

A potentially more serious challenge is that the introduction of the degenerate weights in $u$ and $\underline{u}$ would create terms that cannot be estimated by the energy estimates themselves. Nevertheless, since the weights are chosen to be decreasing toward the future, these uncontrollable terms in fact possess a good sign.

1.4.3. Estimates for the Ricci coefficients

As indicated above, the Ricci coefficients enter as error terms in the energy estimates. Thus, to close all the estimates, we need to control the Ricci coefficients $\Gamma$ by using the transport equations which in turn have the curvature components in the source terms. Since the various Ricci coefficients have different singular behavior, we separate them according to the bounds that they obey. More precisely, denote by $\psi _H$ the components that behave like $f(\underline{u})^{-2}$ as $\underline{u}\to \underline{u}_*$, by $\psi _{\underline{H}}$ the components that behave like $f(u)^{-2}$ as $u\to u_*$, and by $\psi$ the components that are bounded.

For the singular Ricci coefficients $\psi _H$, we have the following schematic transport equations:

$$\nabla _3\psi _H= K+\nabla \psi +\psi \psi +\psi _{\underline{H}}\psi _H.$$

The first three terms on the right-hand side of this equation are bounded while the last term is singular. Nevertheless, the singularity of $\psi _{\underline{H}}$ still allows it to be controlled in $L^1$ along the $e_3$ direction. Thus, this equation can be integrated to show that the initial (singular) bounds for $\psi _H$ can be propagated. It is important that the terms of the form $\psi _H\psi _H$ and $\psi _{\underline{H}}\psi _{\underline{H}}$ do not appear in the equations. A similar structure can also be seen in the equation for the other singular Ricci coefficients $\psi _{\underline{H}}$, which takes the form

$$\nabla _4\psi _{\underline{H}}= K+\nabla \psi +\psi \psi +\psi _{\underline{H}}\psi _H.$$

For the regular Ricci coefficients $\psi$, we have transport equations of the form

$$\nabla _4\psi =\beta +\psi \psi _H\quad \text{or}\quad \nabla _3\psi =\underline{\beta }+\psi \psi _{\underline{H}}.$$

The bounds that we prove show that the right-hand side is integrable, and therefore $\psi$ remains bounded. For example, in the $\nabla _4$ equation, it is important that we do not have terms of the form $\psi _H\psi _H$, $\psi \psi _{\underline{H}}$, $\psi _H\psi _{\underline{H}}$, and $\psi _{\underline{H}}\psi _{\underline{H}}$, which are not uniformly bounded after integrating along the $e_4$ direction.

1.4.4. Null structure in the energy estimates

A priori, the degenerate $L^2$ estimates that we introduce may not be sufficient to control the error terms. Nevertheless, the vacuum Einstein equations possess a remarkable null structure which allows one to close the estimates using only the degenerate $L^2$ estimates.

For example, in the energy estimates for the singular component $\beta$, we have

$$\left\|f(\underline{u})\beta \right\|_{L^2(H)}^2\leq \text{ Data }+ \left\|f^2(\underline{u})\left(\beta \psi _{\underline{H}}\beta +\beta \psi _{H}\underline{\beta }+\beta \psi K\right)\right\|_{L^1_u L^1_{\underline{u}} L^1(S)}.$$

To estimate the first term, it suffices to note that $\psi _{\underline{H}}$, while singular, can be shown to be small after integrating along the $u$ direction. Thus, the first term can be controlled using Gronwall’s inequality. For the second term, since the singularity for $\beta$ has the same strength as that for $\psi _H$ (and similarly the singularity for $\underline{\beta }$ has the same strength as that for $\psi _{\underline{H}}$), the singularity in this term is similar to that in the first term and can also be bounded. The final term is less singular since $\psi$ and $K$ are both uniformly bounded.⁠¹⁵ Notice that if other combinations of curvature terms and Ricci coefficients such as $\beta \psi _H\beta$, $\beta \psi _{\underline{H}}\underline{\beta }$, or $\beta \psi _H K$ appear in the error terms, the degenerate energy will not be strong enough to close the bounds!

¹⁵

Although, as pointed out before, the highest derivative estimates for $K$ in the energy norm suffer a loss as one approaches the singular boundaries, this term can nevertheless be controlled.

✖

In order to close all the estimates, we need to commute also with higher derivatives. As in 1819, we will only commute with angular covariant derivatives. These commutations will not introduce terms that are more singular. Moreover, the null structure of the estimates indicated above is also preserved under these commutations.

Similar to 1819, the renormalization introduces error terms in the energy estimates such that the Ricci coefficients have one more derivative compared to the curvature components. These terms cannot be estimated via transport equations alone but are controlled using also elliptic estimates on the spheres. A form of null structure similar to that described above also makes an appearance in these elliptic estimates, allowing all the bounds to be closed.

1.5. Outline of the paper

We end the introduction with an outline of the remainder of the paper. In section 2, we introduce the basic setup of the paper, including the double null foliation, the coordinate system, and the Einstein vacuum equations recast in terms of the geometric quantities associated to the double null foliation. In section 3, we introduce the norms used in the paper and state a theorem on a priori estimates (Theorem 5) which imply our main existence theorem (Theorem 2). In section 4, we construct a class of characteristic initial data satisfying the assumptions of Theorem 2. In sections 5–8, we prove Theorem 5. In section 5, we obtain the estimates for the metric components and derive functional inequalities useful in our setting. Then in sections 6 and 7, we prove bounds for the Ricci coefficients assuming control of the curvature components. In section 8, we close all the estimates by obtaining bounds for the curvature components. Finally, in section 9, we discuss the nature of the singular boundary and prove Theorems 3 and 4.

2. Basic setup

2.1. Double null foliation

For a smooth⁠¹⁶ spacetime in a neighborhood of $S_{0,0}$, we define a double null foliation as follows: Let $u$ and $\underline{u}$ be solutions to the eikonal equation

¹⁶

The spacetimes considered in this paper are not smooth at $u=u_*$ or $\underline{u}=\underline{u}_*$. However, since we first construct the spacetime in the region $\{u<u_*\}\cap \{\underline{u}<\underline{u}_*\}$ in which the spacetime is smooth (see Theorem 2), it suffices to define the double null foliation for smooth spacetimes.

✖

$$g^{\mu \nu }\partial _\mu u\partial _\nu u=0,\qquad g^{\mu \nu }\partial _\mu \underline{u}\partial _\nu \underline{u}=0,$$

such that $u=0$ on $H_0$ and $\underline{u}=0$ on $\underline{H}_0$. Let

$$L'^\mu =-2g^{\mu \nu }\partial _\nu u,\qquad \underline{L}'^\mu =-2g^{\mu \nu }\partial _\nu \underline{u}.$$

These are null and geodesic vector fields. Let

$$2\Omega ^{-2}=-g\left(L',\underline{L}'\right).$$

Define

$$e_3=\Omega \underline{L}'\text{, }\qquad e_4=\Omega L'$$

to be the normalized null pair such that

$$g(e_3,e_4)=-2$$

and

$$\underline{L}=\Omega ^2\underline{L}'\text{, }L=\Omega ^2 L'$$

to be the so-called equivariant vector fields.

In this paper, we will consider spacetime solutions to the vacuum Einstein equations 1 in the gauge such that

$$\Omega =1,\quad \text{on $H_0$ and $\underline{H}_0$}.$$

The level sets of $u$ (resp. $\underline{u}$) are denoted by $H_u$ (resp. $\underline{H}_{\underline{u}}$). The eikonal equations imply that $H_u$ and $\underline{H}_{\underline{u}}$ are null hypersurfaces. The intersections of the hypersurfaces $H_u$ and $\underline{H}_{\underline{u}}$ are topologically 2-spheres, which we denote by $S_{u,\underline{u}}$. Note that the integral flows of $L$ and $\underline{L}$ respect the foliation $S_{u,\underline{u}}$.

2.2. The coordinate system

We define a coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$ in a neighborhood of $S_{0,0}$ as follows. On the sphere $S_{0,0}$, we have an atlas such that in the local coordinate system $(\theta ^1,\theta ^2)$ in each coordinate chart, the metric $\gamma$ is smooth, bounded, and positive definite. Recall that in a neighborhood of $S_{0,0}$, $u$ and $\underline{u}$ are solutions to the eikonal equations,

$$g^{\mu \nu }\partial _\mu u\partial _\nu u=0,\qquad g^{\mu \nu }\partial _\mu \underline{u}\partial _\nu \underline{u}=0.$$

We then require the coordinates to satisfy

$${\mathcal{L}} \mkern -9mu /\,_{\underline{L}} \theta ^A=0$$

on the initial hypersurface $\underline{H}_0$ and

$${\mathcal{L}} \mkern -9mu /\,_L \theta ^A=0$$

in the spacetime region. Here, ${\mathcal{L}} \mkern -9mu /\,_L$ and ${\mathcal{L}} \mkern -9mu /\,_{\underline{L}}$ denote the restriction of the Lie derivative to $TS_{u,\underline{u}}$ (See 5, Chapter 1.) and $L$ and $\underline{L}$ are defined as in section 2.1. Relative to the coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$, the null pair $e_3$ and $e_4$ can be expressed as

$$e_3=\Omega ^{-1}\left(\frac{\partial }{\partial u}+b^A\frac{\partial }{\partial \theta ^A}\right),\qquad e_4=\Omega ^{-1}\frac{\partial }{\partial \underline{u}},$$

for some $b^A$ such that $b^A=0$ on $\underline{H}_0$, while the metric $g$ takes the form

$$g=-2\Omega ^2\left(du\otimes d\underline{u}+d\underline{u}\otimes du\right)+\gamma _{AB}\left(d\theta ^A-b^Adu\right)\otimes \left(d\theta ^B-b^Bdu\right).$$

2.3. Equations

We will recast the Einstein equations as a system for Ricci coefficients and curvature components associated to a null frame $e_3$, $e_4$ defined above and an orthonormal frame⁠¹⁷ $\{e_A\}_{A=1,2}$ tangent to the 2-spheres $S_{u,\underline{u}}$. We define the Ricci coefficients relative to the null fame,

¹⁷

Of course the orthonormal frame is only defined locally. Alternatively, the capital Latin indices can be understood as abstract indices.

✖

$$\begin{equation} \begin{split} \chi _{AB}&=g\left(D_A e_4,e_B\right), \qquad \ \underline{\chi }_{AB}=g\left(D_A e_3,e_B\right),\\ \eta _A&=-\frac{1}{2} g\left(D_3 e_A,e_4\right), \qquad \underline{\eta }_A=-\frac{1}{2} g\left(D_4 e_A,e_3\right){,}\\ \omega &=-\frac{1}{4} g\left(D_4 e_3,e_4\right), \qquad \ \ \underline{\omega }=-\frac{1}{4} g\left(D_3 e_4,e_3\right),\\ \zeta _A&=\frac{1}{2} g\left(D_A e_4,e_3\right), \end{split} \tag{5} \end{equation}$$

where $D_A=D_{e_{(A)}}$. We also introduce the null curvature components,

$$\begin{equation} \begin{split} \alpha _{AB}&=R\left(e_A, e_4, e_B, e_4\right),\qquad \underline{\alpha }_{AB}=R\left(e_A, e_3, e_B, e_3\right),\\ \beta _A&= \frac{1}{2} R\left(e_A, e_4, e_3, e_4\right),\qquad \underline{\beta }_A =\frac{1}{2} R\left(e_A, e_3, e_3, e_4\right),\\ \rho &=\frac{1}{4} R\left(e_4,e_3, e_4, e_3\right), \qquad \sigma =\frac{1}{4} \,^*R\left(e_4,e_3, e_4, e_3\right). \end{split} \tag{6} \end{equation}$$

Here $\, ^*R$ denotes the Hodge dual of $R$. We denote by $\nabla$ the induced covariant derivative operator on $S_{u,\underline{u}}$ and by $\nabla _3$, $\nabla _4$ the projections to $S_{u,\underline{u}}$ of the covariant derivatives $D_3$, $D_4$ (see precise definitions in 15, Chapter 3.1).

Observe that

$$\begin{equation} \begin{split} &\omega =-\frac{1}{2} \nabla _4 (\log \Omega ),\qquad \underline{\omega }=-\frac{1}{2} \nabla _3 (\log \Omega ),\\ &\eta _A=\zeta _A +\nabla _A (\log \Omega ),\quad \underline{\eta }_A=-\zeta _A+\nabla _A (\log \Omega ). \end{split} {\tag{7}} \end{equation}$$

Define the following contractions of the tensor product $\phi ^{(1)}$ and $\phi ^{(2)}$ with respect to the metric $\gamma$:

$$\phi ^{(1)}\cdot \phi ^{(2)}:=(\gamma ^{-1})^{AC}(\gamma ^{-1})^{BD}\phi ^{(1)}_{AB}\phi ^{(2)}_{CD} \quad \text{for symmetric $2$-tensors $\phi ^{(1)}_{AB}$, $\phi ^{(2)}_{AB}$,}$$

$$\phi ^{(1)}\cdot \phi ^{(2)}:=(\gamma ^{-1})^{AB}\phi ^{(1)}_{A}\phi ^{(2)}_{B} \quad \text{for $1$-forms $\phi ^{(1)}_{A}$, $\phi ^{(2)}_{A}$,}$$

$$(\phi ^{(1)}\cdot \phi ^{(2)})_A:=(\gamma ^{-1})^{BC}\phi ^{(1)}_{AB}\phi ^{(2)}_{C} \quad \text{for a symmetric $2$-tensor $\phi ^{(1)}_{AB}$ and a $1$-form $\phi ^{(2)}_{A}$,}$$

$$(\phi ^{(1)}\widehat{\otimes }\phi ^{(2)})_{AB}:=\phi ^{(1)}_A\phi ^{(2)}_B+\phi ^{(1)}_B\phi ^{(2)}_A-{\gamma _{AB}}(\phi ^{(1)}\cdot \phi ^{(2)}) \quad \text{for $1$-forms $\phi ^{(1)}_A$, $\phi ^{(2)}_A$,}$$

$$\phi ^{(1)}\wedge \phi ^{(2)}:={\epsilon } \mkern -7mu /\,^{AB}(\gamma ^{-1})^{CD}\phi ^{(1)}_{AC}\phi ^{(2)}_{BD}\quad \text{for symmetric $2$-tensors $\phi ^{(1)}_{AB}$, $\phi ^{(2)}_{AB}$},$$

where ${\epsilon } \mkern -7mu /\,$ is the volume form associated to the metric $\gamma$. We also define by $^*$ for $1$-forms and symmetric $2$-tensors, respectively, as follows (note that on $1$-forms this is the Hodge dual on $S_{u,\underline{u}}$):

$$\begin{align*} ^*\phi _A := \,& \gamma _{AC} {\epsilon } \mkern -7mu /\,^{CB} \phi _B, \\ ^*\phi _{AB} :=\, & \gamma _{BD} {\epsilon } \mkern -7mu /\,^{DC} \phi _{AC}. \end{align*}$$

Define the operator $\nabla \widehat{\otimes }$ on a $1$-form $\phi _{A}$ by

$$\left(\nabla \widehat{\otimes }\phi \right)_{AB} := \nabla _A \phi _B + \nabla _B \phi _A - \gamma _{AB} \mathrm{div\,}\phi .$$

For totally symmetric tensors, define the $\mathrm{div\,}$ and $\mathrm{curl\,}$ operators as follows

$$\left(\mathrm{div\,}\phi \right)_{A_1\cdots A_r}:=\nabla ^B\phi _{BA_1\cdots A_r},$$

$$\left(\mathrm{curl\,}\phi \right)_{A_1\cdots A_r}:={\epsilon } \mkern -7mu /\,^{BC}\nabla _B\phi _{CA_1\cdots A_r}.$$ Define also the trace of totally symmetric tensors to be

$$\left(\mathrm{tr\,}\phi \right)_{A_1\cdots A_{r-1}}:=\left(\gamma ^{-1}\right)^{BC}\phi _{BCA_1\cdots A_{r-1}}.{}$$

We separate the trace and traceless part of $\chi$ and $\underline{\chi }$. Let $\hat{\chi }$ and $\hat{\underline{\chi }}$ be the traceless parts of $\chi$ and $\underline{\chi }$, respectively. Then $\chi$ and $\underline{\chi }$ satisfy the following null structure equations:

$$\begin{equation} \begin{split} \nabla _4 \mathrm{tr\,}\chi +\frac{1}{2} (\mathrm{tr\,}\chi )^2&=-|\hat{\chi }|^2-2\omega \mathrm{tr\,}\chi {,}\\ \nabla _4\hat{\chi }+\mathrm{tr\,}\chi \hat{\chi }&=-2 \omega \hat{\chi }-\alpha {,}\\ \nabla _3 \mathrm{tr\,}\underline{\chi }+\frac{1}{2} (\mathrm{tr\,}\underline{\chi })^2&=-2\underline{\omega }\mathrm{tr\,}\underline{\chi }-|\hat{\underline{\chi }}|^2{,}\\ \nabla _3\hat{\underline{\chi }}+ \mathrm{tr\,}\underline{\chi }\, \hat{\underline{\chi }}&= -2\underline{\omega }\hat{\underline{\chi }}-\underline{\alpha }{,}\\ \nabla _4 \mathrm{tr\,}\underline{\chi }+\frac{1}{2} \mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi }&=2\omega \mathrm{tr\,}\underline{\chi }+2\rho - \hat{\chi }\cdot \hat{\underline{\chi }}+2\mathrm{div\,}\underline{\eta }+2|\underline{\eta }|^2{,}\\ \nabla _4\hat{\underline{\chi }}+\frac{1}{2} \mathrm{tr\,}\chi \hat{\underline{\chi }}&=\nabla \widehat{\otimes } \underline{\eta }+2\omega \hat{\underline{\chi }}-\frac{1}{2} \mathrm{tr\,}\underline{\chi }\hat{\chi }+\underline{\eta }\widehat{\otimes } \underline{\eta }{,}\\ \nabla _3 \mathrm{tr\,}\chi +\frac{1}{2} \mathrm{tr\,}\underline{\chi }\mathrm{tr\,}\chi &=2\underline{\omega }\mathrm{tr\,}\chi +2\rho - \hat{\chi }\cdot \hat{\underline{\chi }}+2\mathrm{div\,}\eta +2|\eta |^2{,}\\ \nabla _3\hat{\chi }+\frac{1}{2} \mathrm{tr\,}\underline{\chi }\hat{\chi }&=\nabla \widehat{\otimes } \eta +2\underline{\omega }\hat{\chi }-\frac{1}{2} \mathrm{tr\,}\chi \hat{\underline{\chi }}+\eta \widehat{\otimes } \eta {.} \end{split} {\tag{8}} \end{equation}$$

The other Ricci coefficients satisfy the following null structure equations:

$$\begin{equation} \begin{split} \nabla _4\eta &=-\chi \cdot \left(\eta -\underline{\eta }\right)-\beta {,}\\ \nabla _3\underline{\eta }&=-\underline{\chi }\cdot \left(\underline{\eta }-\eta \right)+\underline{\beta }{,}\\ \nabla _4\underline{\omega }&=2\omega \underline{\omega }-\eta \cdot \underline{\eta }+\frac{1}{2}\left|\eta \right|^2+\frac{1}{2} \rho ,\\ \nabla _3\omega &=2\omega \underline{\omega }-\eta \cdot \underline{\eta }+\frac{1}{2}\left|\underline{\eta }\right|^2+\frac{1}{2} \rho . \end{split} {\tag{9}} \end{equation}$$

The Ricci coefficients also satisfy the following constraint equations:

$$\begin{equation} \begin{split} \mathrm{div\,}\hat{\chi }&=\frac{1}{2} \nabla \mathrm{tr\,}\chi - \frac{1}{2} \left(\eta -\underline{\eta }\right)\cdot \left(\hat{\chi }-\frac{1}{2} \mathrm{tr\,}\chi \right) -\beta ,\\ \mathrm{div\,}\hat{\underline{\chi }}&=\frac{1}{2} \nabla \mathrm{tr\,}\underline{\chi }+ \frac{1}{2} \left(\eta -\underline{\eta }\right)\cdot \left(\hat{\underline{\chi }}-\frac{1}{2} \mathrm{tr\,}\underline{\chi }\right) +\underline{\beta }{,}\\ \mathrm{curl\,}\eta &=-\mathrm{curl\,}\underline{\eta }=\sigma +\frac{1}{2}\hat{\underline{\chi }}\wedge \hat{\chi }{,}\\ K&=-\rho +\frac{1}{2} \hat{\chi }\cdot \hat{\underline{\chi }}-\frac{1}{4} \mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi }{,} \end{split} {\tag{10}} \end{equation}$$

with $K$ the Gauss curvature of the spheres $S_{u,\underline{u}}$. The null curvature components satisfy the following null Bianchi equations:

$$\begin{equation} \begin{split} \nabla _3\alpha +\frac{1}{2} \mathrm{tr\,}\underline{\chi }\alpha &={\nabla }\widehat{\otimes }\beta + 4\underline{\omega }\alpha -3\left(\hat{\chi }\rho +^*\hat{\chi }\sigma \right)+ \left(\zeta +4\eta \right)\widehat{\otimes }\beta ,\\ \nabla _4\beta +2\mathrm{tr\,}\chi \beta &= \mathrm{div\,}\alpha - 2\omega \beta + \left(2\zeta +\underline{\eta }\right)\cdot \alpha ,\\ \nabla _3\beta +\mathrm{tr\,}\underline{\chi }\beta &={\nabla }\rho + 2\underline{\omega }\beta +^*{\nabla }\sigma +2\hat{\chi }\cdot \underline{\beta }+3\left(\eta \rho +^*\eta \sigma \right),\\ \nabla _4\sigma +\frac{3}{2}\mathrm{tr\,}\chi \sigma &=-\mathrm{div\,}^*\beta +\frac{1}{2}\hat{\underline{\chi }}\wedge \alpha -\zeta \wedge \beta -2\underline{\eta }\wedge \beta ,\\ \nabla _3\sigma +\frac{3}{2}\mathrm{tr\,}\underline{\chi }\sigma &=-\mathrm{div\,}^*\underline{\beta }-\frac{1}{2}\hat{\chi }\wedge \underline{\alpha }+\zeta \wedge \underline{\beta }-2\eta \wedge \underline{\beta },\\ \nabla _4\rho +\frac{3}{2}\mathrm{tr\,}\chi \rho &=\mathrm{div\,}\beta -\frac{1}{2}\hat{\underline{\chi }}\cdot \alpha +\zeta \cdot \beta +2\underline{\eta }\cdot \beta ,\\ \nabla _3\rho +\frac{3}{2}\mathrm{tr\,}\underline{\chi }\rho &=-\mathrm{div\,}\underline{\beta }- \frac{1}{2}\hat{\chi }\cdot \underline{\alpha }+\zeta \cdot \underline{\beta }-2\eta \cdot \underline{\beta },\\ \nabla _4\underline{\beta }+\mathrm{tr\,}\chi \underline{\beta }&=-{\nabla }\rho +^*{\nabla }\sigma + 2\omega \underline{\beta }+2\hat{\underline{\chi }}\cdot \beta -3\left(\underline{\eta }\rho -^*\underline{\eta }\sigma \right),\\ \nabla _3\underline{\beta }+2\mathrm{tr\,}\underline{\chi }\underline{\beta }&=-\mathrm{div\,}\underline{\alpha }-2\underline{\omega }\underline{\beta }-\left(-2\zeta +\eta \right) \cdot \underline{\alpha },\\ \nabla _4\underline{\alpha }+\frac{1}{2} \mathrm{tr\,}\chi \underline{\alpha }&=-{\nabla }\widehat{\otimes }\underline{\beta }+ 4\omega \underline{\alpha }-3\left(\hat{\underline{\chi }}\rho -^*\hat{\underline{\chi }}\sigma \right)+ (\zeta -4\underline{\eta })\widehat{\otimes }\underline{\beta }, \end{split} {\tag{11}} \end{equation}$$

where $^*$ denotes the Hodge dual on $S_{u,\underline{u}}$.

We now rewrite the Bianchi equations in terms of the Gauss curvature $K$ of the spheres $S_{u,\underline{u}}$ and the renormalized curvature component $\check{\sigma }$ defined by

$$\check{\sigma }=\sigma +\frac{1}{2} \hat{\underline{\chi }}\wedge \hat{\chi }.$$

The Bianchi equations take the following form:

$$\begin{equation} \begin{split} \nabla _3\beta +\mathrm{tr\,}\underline{\chi }\beta =&-\nabla K +^*{\nabla }\check{\sigma }+ 2\underline{\omega }\beta +2\hat{\chi }\cdot \underline{\beta }-3\left(\eta K-^*\eta \check{\sigma }\right)\\ &+\frac{1}{2}\left({\nabla }\left(\hat{\chi }\cdot \hat{\underline{\chi }}\right)+^*{\nabla }\left(\hat{\chi }\wedge \hat{\underline{\chi }}\right)\right)+\frac{3}{2}\left(\eta \hat{\chi }\cdot \hat{\underline{\chi }}+^*\eta \hat{\chi }\wedge \hat{\underline{\chi }}\right)\\ &-\frac{1}{4} \left(\nabla \mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi }+\mathrm{tr\,}\chi \nabla \mathrm{tr\,}\underline{\chi }\right)-\frac{3}{4} \eta \mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi },\\ \nabla _4\check{\sigma }+\frac{3}{2}\mathrm{tr\,}\chi \check{\sigma }=&-\mathrm{div\,}^*\beta -\zeta \wedge \beta -2\underline{\eta }\wedge \beta -\frac{1}{2} \hat{\chi }\wedge \left(\nabla \widehat{\otimes }\underline{\eta }\right)-\frac{1}{2} \hat{\chi }\wedge \left(\underline{\eta }\widehat{\otimes }\underline{\eta }\right),\\ \nabla _4 K+\mathrm{tr\,}\chi K=&-\mathrm{div\,}\beta -\zeta \cdot \beta -2\underline{\eta }\cdot \beta +\frac{1}{2} \hat{\chi }\cdot \nabla \widehat{\otimes }\underline{\eta }+\frac{1}{2} \hat{\chi }\cdot \left(\underline{\eta }\widehat{\otimes }\underline{\eta }\right)\\ &-\frac{1}{2} \mathrm{tr\,}\chi \mathrm{div\,}\underline{\eta }-\frac{1}{2}\mathrm{tr\,}\chi |\underline{\eta }|^2,\\ \nabla _3\check{\sigma }+\frac{3}{2}\mathrm{tr\,}\underline{\chi }\check{\sigma }=&-\mathrm{div\,}^*\underline{\beta }+\zeta \wedge \underline{\beta }-2\eta \wedge \underline{\beta }+\frac{1}{2} \hat{\underline{\chi }}\wedge \left(\nabla \widehat{\otimes }\eta \right)+\frac{1}{2} \hat{\underline{\chi }}\wedge \left(\eta \widehat{\otimes }\eta \right),\\ \nabla _3 K+\mathrm{tr\,}\underline{\chi }K=&\ \mathrm{div\,}\underline{\beta }-\zeta \cdot \underline{\beta }+2\eta \cdot \underline{\beta }+\frac{1}{2} \hat{\underline{\chi }}\cdot \nabla \widehat{\otimes }\eta +\frac{1}{2} \hat{\underline{\chi }}\cdot \left(\eta \widehat{\otimes }\eta \right)\\ &-\frac{1}{2} \mathrm{tr\,}\underline{\chi }\mathrm{div\,}\eta -\frac{1}{2} \mathrm{tr\,}\underline{\chi }|\eta |^2,\\ \nabla _4\underline{\beta }+\mathrm{tr\,}\chi \underline{\beta }=&\ {\nabla } K +^*{\nabla }\check{\sigma }+ 2\omega \underline{\beta }+2\hat{\underline{\chi }}\cdot \beta +3\left(\underline{\eta }K+^*\underline{\eta }\check{\sigma }\right)\\ &-\frac{1}{2}\left({\nabla }\left(\hat{\chi }\cdot \hat{\underline{\chi }}\right)-^*{\nabla }\left(\hat{\chi }\wedge \hat{\underline{\chi }}\right)\right)-\frac{3}{2}\left(\underline{\eta }\hat{\chi }\cdot \hat{\underline{\chi }}-^*\underline{\eta }\hat{\chi }\wedge \hat{\underline{\chi }}\right)\\ &+\frac{1}{4} \left(\nabla \mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi }+\mathrm{tr\,}\chi \nabla \mathrm{tr\,}\underline{\chi }\right)+\frac{3}{4} \underline{\eta }\mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi }. \end{split} {\tag{12}} \end{equation}$$

Notice that we have obtained a system for the renormalized curvature components in which the curvature components $\alpha$ and $\underline{\alpha }$ do not appear.⁠¹⁸

¹⁸

Moreover, compared to the renormalization in 19, this system does not contain the terms $\mathrm{tr\,}\chi |\hat{\underline{\chi }}|^2$ and $\mathrm{tr\,}\underline{\chi }|\hat{\chi }|^2$, which would be uncontrollable in the context of this paper.

✖

From now on, we will use capital Latin letters $A\in \{1,2\}$ for indices on the spheres $S_{u,\underline{u}}$ and Greek letters $\mu \in \{1,2,3,4\}$ for indices in the whole spacetime.

2.4. Schematic notation

We define a schematic notation for the Ricci coefficients according to the estimates that they obey. Introduce the following conventions:⁠¹⁹

¹⁹

Notice that this definition is different form that in 19, since in the context of the present paper $\mathrm{tr\,}\chi$ and $\mathrm{tr\,}\underline{\chi }$ verify different bounds compared to 19.

✖

$$\psi \in \left\{\eta ,\underline{\eta }\right\},\quad \psi _H\in \left\{\mathrm{tr\,}\chi ,\hat{\chi },\omega \right\},\quad \psi _{\underline{H}}\in \left\{\mathrm{tr\,}\underline{\chi },\hat{\underline{\chi }},\underline{\omega }\right\}.$$

We will use this schematic notation only in the situations where the exact constant in front of the term is irrelevant to the argument. We will denote by $\psi \psi$ (or $\psi \psi _H$, etc.) an arbitrary contraction with respect to the metric $\gamma$ and by $\nabla \psi$ an arbitrary angular covariant derivative. $\nabla ^i\psi ^j$ will be used to denote the sum of all terms which are products of $j$ factors, such that each factor takes the form $\nabla ^{i_k}\psi$ and that the sum of all $i_k$’s is $i$, i.e.,

$$\nabla ^i\psi ^j=\displaystyle \sum _{i_1+i_2+\cdots +i_j{=i}}\underbrace{\nabla ^{i_1}\psi \nabla ^{i_2}\psi \cdots \nabla ^{i_j}\psi }_\text{j factors}.$$

We will use brackets to denote terms with one of the components in the brackets. For instance, the notation $\psi (\psi ,\psi _H)$ denotes the sum of all terms of the form $\psi \psi$ or $\psi \psi _H$.

In this schematic notation, the Ricci coefficients $\psi _H$ satisfy

$$\nabla _3 \psi _H = K+ \nabla \psi + \psi \psi +\psi _H \psi _{\underline{H}}.$$

The Ricci coefficients $\psi _{\underline{H}}$ similarly obey

$$\nabla _4 \psi _{\underline{H}} = K+ \nabla \psi + \psi \psi +\psi _H \psi _{\underline{H}}.$$

The Ricci coefficients $\psi$ obey either one of the following equations:

$$\nabla _3 \psi = \underline{\beta }+ \psi \psi _{\underline{H}}$$

or

$$\nabla _4 \psi = \beta + \psi \psi _H.$$

We also rewrite the Bianchi equations in the following schematic notation:

$$\begin{equation} \begin{split} \nabla _3\beta {+}\nabla K -^*\nabla \check{\sigma }=&\ \sum _{i_1+i_2=1}\psi _{\underline{H}}\psi ^{i_1} \nabla ^{i_2}\psi _H+\psi K+\sum _{i_1+i_2=1} \psi ^{i_1}\psi \nabla ^{i_2}\psi ,\\ \nabla _4\check{\sigma }+\mathrm{div\,}^*\beta =&\ \psi _H\check{\sigma }+\psi \sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _H,\\ \nabla _4 K{+}\mathrm{div\,}\beta =\,&\psi _H K+\psi \sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _H,\\ \nabla _3\check{\sigma }+\mathrm{div\,}^*\underline{\beta }=&\ \psi _{\underline{H}}\check{\sigma }+\psi \sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _{\underline{H}},\\ \nabla _3 K{-}\mathrm{div\,}\underline{\beta }=&\ \psi _{\underline{H}}K+\psi \sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _{\underline{H}},\\ \nabla _4\underline{\beta }{-}\nabla K -^*\nabla \check{\sigma }=&\ \sum _{i_1+i_2=1}\psi _{H}\psi ^{i_1} \nabla ^{i_2}\psi _{\underline{H}}+\psi K+\sum _{i_1+i_2=1} \psi ^{i_1}\psi \nabla ^{i_2}\psi . \end{split} {\tag{13}} \end{equation}$$

3. Norms

In this section we define the norms that we will use to control the geometric quantities. We will in particular use the schematic notation defined in section 2.4. Our norms will be of the form $L^p_u L^q_{\underline{u}} L^r(S)$, where $L^p_u$ and $L^q_{\underline{u}}$ are defined with respect to the measures $du$ and $d\underline{u}$, respectively, and $L^r(S)$ is defined for any tensors $\phi$ on $S_{u,\underline{u}}$ by

$$\|\phi \|_{L^r(S_{u,\underline{u}})}:=\left(\int _{S_{u,\underline{u}}} \left(\phi _{A_1 A_2 \cdots A_n} \phi ^{A_1 A_2 \cdots A_n}\right)^{\frac{r}{2}}\right)^{\frac{1}{r}},$$

where the integral is with respect to the volume form induced by $\gamma$.

We define the following norms for the Ricci coefficients $\psi$ for $p\in [1,\infty ]$, $i\in \mathbb{N}$:

$$\begin{equation} \mathcal{O}_{i,p}\left[\psi \right]:=\left\|\nabla ^i\psi \right\|_{L^\infty _uL^\infty _{\underline{u}}L^p(S)}. {\tag{14}} \end{equation}$$

Define the following norms for the Ricci coefficients $\psi _H$ for $p\in [1,\infty ]$, $i\in \mathbb{N}$:

$$\begin{equation} \mathcal{O}_{i,p}\left[\psi _H\right]:=\left\|f(\underline{u})\nabla ^i\psi _H\right\|_{L^2_{\underline{u}}L^\infty _uL^p(S)}. {\tag{15}} \end{equation}$$

Similarly, we define the following norms for the Ricci coefficients $\psi _{\underline{H}}$ for $p\in [1,\infty ]$, $i\in \mathbb{N}$:

$$\begin{equation} \mathcal{O}_{i,p}\left[\psi _{\underline{H}}\right]:=\left\|f(u)\nabla ^i\psi _{\underline{H}}\right\|_{L^2_{u}L^\infty _{\underline{u}}L^p(S)}. {\tag{16}} \end{equation}$$

As a shorthand, we define the following norm combining all of the norms above:

$$\mathcal{O}_{i,p}:=\sum _{\psi \in \{\eta ,\underline{\eta }\}}\mathcal{O}_{i,p}\left[\psi \right]+\sum _{\psi _H\in \{tr\chi ,\hat{\chi },\omega \}}\mathcal{O}_{i,p}\left[\psi _H\right]+\sum _{\psi _{\underline{H}}\in \{tr \underline{\chi },\hat{\underline{\chi }},\underline{\omega }\}}\mathcal{O}_{i,p}\left[\psi _{\underline{H}}\right].$$

We make two remarks concerning these norms.

Remark 7.

While the norms for $\psi _H$ and $\psi _{\underline{H}}$ are based on $L^2$ in $\underline{u}$ and $u$, respectively, by virtue of the weights $f(\underline{u})$ and $f(u)$, they actually control the $L^1$ norms. More precisely, since $\int _0^{\underline{u}_*} \frac{1}{f^2(\underline{u}')} d\underline{u}' <\epsilon ^{{2}}$ and $\int _0^{u_*} \frac{1}{f^2(u')} du' <\epsilon ^{{2}}$, by the Cauchy–Schwarz inequality we have

$$\left\|\nabla ^i\psi _H\right\|_{L^1_{\underline{u}}L^\infty _u L^p(S)}\leq C\epsilon \mathcal{O}_{i,p}\left[\psi _H\right]$$

and

$$\left\|\nabla ^i\psi _{\underline{H}}\right\|_{L^1_{u}L^\infty _{\underline{u}} L^p(S)}\leq C\epsilon \mathcal{O}_{i,p}\left[\psi _{\underline{H}}\right].$$

Remark 8.

The norm $\mathcal{O}_{i,p}[\psi _H]$ (resp. $\mathcal{O}_{i,p}[\psi _{\underline{H}}]$) allows us to first take $L^\infty$ along the $u$ direction (resp. $\underline{u}$ direction) before the $L^2$ norm in $\underline{u}$ (resp. $u$) is taken. This is stronger than the norms such that the order is reversed; i.e., we have

$$\left\|f(\underline{u})\nabla ^i\psi _H\right\|_{L^\infty _uL^2_{\underline{u}} L^p(S)}\leq C \mathcal{O}_{i,p}\left[\psi _H\right]$$

and

$$\left\|f(u)\nabla ^i\psi _{\underline{H}}\right\|_{L^\infty _{\underline{u}}L^2_{\underline{u}} L^p(S)}\leq C \mathcal{O}_{i,p}\left[\psi _{\underline{H}}\right].$$

In addition to the above norms, we need to define norms for the highest derivatives for the Ricci coefficients. Let

$$\begin{equation} \begin{split} \tilde{\mathcal{O}}_{4,2}:=&\ \left\|f(\underline{u})^2\nabla ^4\mathrm{tr\,}\chi \right\|_{L^\infty _u L^\infty _{\underline{u}}L^2(S)}+\left\|f(u)^2\nabla ^4\mathrm{tr\,}\underline{\chi }\right\|_{L^\infty _u L^\infty _{\underline{u}}L^2(S)}\\ &+\left\|f(\underline{u})\nabla ^4(\hat{\chi },\omega )\right\|_{L^\infty _u L^2_{\underline{u}}L^2(S)}+\left\|f(u)\nabla ^4(\eta ,\underline{\eta })\right\|_{L^\infty _u L^2_{\underline{u}}L^2(S)}\\ &+\left\|f(u)\nabla ^4(\hat{\underline{\chi }},\underline{\omega })\right\|_{L^\infty _{\underline{u}} L^2_uL^2(S)}+\left\|f(\underline{u})\nabla ^4(\eta ,\underline{\eta })\right\|_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}. \end{split} {\tag{17}} \end{equation}$$

Remark 9.

Here, note that for the norms for $\hat{\chi }$, $\omega$, $\eta$, $\underline{\eta }$, $\hat{\underline{\chi }}$, and $\omega$, $L^\infty$ in $\underline{u}$ (or $u$) is taken after $L^2$ in $u$ (or $\underline{u}$). According to Remark 8, this is weaker than the $\mathcal{O}_{i,2}$ norms defined above.

Remark 10.

Notice that the norms for the fourth derivatives of $\eta$ and $\underline{\eta }$ come with a weight $f(u)$ or $f(\underline{u})$. This is in contrast to the lower-order derivatives for $\eta$ and $\underline{\eta }$, which can be estimated in $L^\infty _uL^\infty _{\underline{u}}$ without any degeneration. The degeneration here arises from the fact that these higher-order derivatives are recovered from the energy estimates for $\nabla ^3 K$. These energy estimates for $\nabla ^3 K$, which are derived simultaneously with the estimates for the singular components $\nabla ^3\beta$ or $\nabla ^3\underline{\beta }$, have a degeneration either in $\underline{u}$ or $u$.

We also define the curvature norms for the curvature components. For $i\in \mathbb{N}$, let

$$\begin{equation} \begin{split} \mathcal{R}_i:=&\ \left\|f(\underline{u})\nabla ^i\beta \right\|_{L^\infty _u L^2_{\underline{u}}L^2(S)}+\left\|f(u)\nabla ^i(K,\check{\sigma })\right\|_{L^\infty _u L^2_{\underline{u}}L^2(S)}\\ &+\left\|f(\underline{u})\nabla ^i(K,\check{\sigma })\right\|_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}+\left\|f(u)\nabla ^i\underline{\beta }\right\|_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}. \end{split} {\tag{18}} \end{equation}$$

As a shorthand, we also let

$$\mathcal{R}:=\sum _{i\leq 3} \mathcal{R}_i.$$

Finally, let $\mathcal{O}_{\mathrm{ini}}$ and $\mathcal{R}_{\mathrm{ini}}$ denote the corresponding norms for the initial data, i.e.,

$$\begin{equation*} \begin{split} \mathcal{O}_{\mathrm{ini}}:=&\sum _{i\leq 3}\left(\left\|\nabla ^i\psi \right\|_{L^\infty _{\underline{u}}L^{2}\left(S_{0,\underline{u}}\right)}+\left\|\nabla ^i\psi \right\|_{L^\infty _{u}L^{2}\left(S_{u,0}\right)}\right.\\ &\left.\qquad +\left\|f(\underline{u})\nabla ^i\psi _H\right\|_{L^2_{\underline{u}}L^{{2}}\left(S_{0,\underline{u}}\right)}+\left\|f(u)\nabla ^i\psi _{\underline{H}}\right\|_{L^2_{u}L^{{2}}\left(S_{u,0}\right)}\right)\\ &+\left\|f(\underline{u})^2\nabla ^4\mathrm{tr\,}\chi \right\|_{L^\infty _{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}+\left\|f(u)^2\nabla ^4\mathrm{tr\,}\underline{\chi }\right\|_{L^\infty _u L^2\left(S_{u,0}\right)}\\ &+\left\|\nabla ^4\mathrm{tr\,}\underline{\chi }\right\|_{L^\infty _{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}+\left\|\nabla ^4\mathrm{tr\,}\chi \right\|_{L^\infty _u L^2\left(S_{u,0}\right)}\\ &+\left\|f(\underline{u})\nabla ^4\left(\hat{\chi },\omega \right)\right\|_{L^2_{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}+\left\|\nabla ^4\left(\eta ,\underline{\eta }\right)\right\|_{L^2_{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}\\ &+\left\|f(u)\nabla ^4\left(\hat{\underline{\chi }},\underline{\omega }\right)\right\|_{L^2_uL^2\left(S_{u,0}\right)}+\left\|\nabla ^4\left(\eta ,\underline{\eta }\right)\right\|_{L^2_{u}L^2(S_{u,0})} \end{split} \end{equation*}$$

and

$$\begin{equation*} \begin{split} \mathcal{R}_{\mathrm{ini}} :=&{\sum _{i\leq 3}}\left(\left\|f(\underline{u})\nabla ^i\beta \right\|_{L^2_{\underline{u}}L^2(S_{0,\underline{u}})}+\left\|\nabla ^i(K,\check{\sigma })\right\|_{L^2_{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}\right.\\ &\left.\qquad +\left\|\nabla ^i\left(K,\check{\sigma }\right)\right\|_{L^2_{u}L^2\left(S_{u,0}\right)}+\left\|f(u)\nabla ^i\underline{\beta }\right\|_{L^2_{u}L^2\left(S_{u,0}\right)}\right). \end{split} \end{equation*}$$

In order to prove Theorem 2, we will establish a priori estimates for the geometric quantities in the above norms:

Theorem 5.

Assume that the initial data for the characteristic initial value problem satisfy the assumptions of Theorem 2 with $\epsilon$ sufficiently small. Then there exists $B$ depending only on $D$ and $d$ such that

$$\sum _{i\leq 3}\mathcal{O}_{i,2}+\tilde{\mathcal{O}}_{4,2}+\mathcal{R}\leq B.$$

In the remainder of the paper, we will focus on the proof of Theorem 5 (after constructing initial data sets in the next section). Standard methods show that Theorem 5 implies Theorem 2. We will omit the details and refer the readers to 519 for a proof that the a priori estimates imply the existence theorem.

Remark 11.

The assumptions of Theorem 2 imply the boundedness of the following weighted $L^2$ norms of the curvature components:

$$\sum _{i\leq 3}\left\|f(\underline{u})\nabla ^i\beta \right\|_{L^2_{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}+\sum _{i\leq 3}\left\|{\nabla ^i(K,\check{\sigma })}\right\|_{L^2_{\underline{u}}L^2\left(S_{0,\underline{u}}\right)}\leq \tilde{D}$$

and

$$\sum _{i\leq 3}\left\|f({u})\nabla ^i\underline{\beta }\right\|_{L^2_{u}L^2\left(S_{u,0}\right)}+\sum _{i\leq 3}\left\|{\nabla ^i\left(K,\check{\sigma }\right)}\right\|_{L^2_{{u}}L^2\left(S_{u,0}\right)}\leq \tilde{D}$$

for some $\tilde{D}$ depending only on $D$ and $d$. These estimates for $\beta$, $\check{\sigma }$, and $\underline{\beta }$ follow immediately from the constraint equations on the 2-spheres (see 10). The bound for $K$ follows after integrating the null Bianchi equations for $K$ on each of the initial null hypersurfaces (see 12).⁠²⁰ In particular the assumptions of Theorem 2 imply that

²⁰

Notice that it is precisely for the initial bound for $K$ that we require an extra derivative for $\chi$ on $H_0$ (and $\underline{\chi }$ on $\underline{H}_0$) in the assumptions of the theorem. This is related to the intrinsic loss of derivatives for the characteristic initial value problem for second-order hyperbolic systems (see 21).

✖

$$\mathcal{O}_{\mathrm{ini}} +\mathcal{R}_{\mathrm{ini}} \leq \tilde{D}.$$

4. Construction of initial data set

In this section we construct initial data sets satisfying the assumptions of Theorems 2 and 4. In particular we show that the constraint equations can be solved for $|\hat{\chi }(0,\underline{u})|\sim (f(\underline{u}))^{-2}$ and $|\hat{\underline{\chi }}(u,0)| \sim (f(u))^{-2}$. Our approach in this section follows closely that of Christodoulou in 5, Chapter 2.

Assume for simplicity that $S_{0,0}$ is a standard sphere of radius $1$. Introduce⁠²¹ the standard stereographic coordinates $(\theta ^1, \theta ^2)$ such that the standard metric $\stackrel{\circ }{\gamma }$ on the sphere takes the form

²¹

While we only write down one coordinate chart, it is implicit that we have two stereographic charts—the north pole chart and the south pole chart. In the following, when we derive the estimates for the geometric quantities, we only prove the bounds in a sufficiently large ball $B_\rho$ in each of these charts.

✖

$$\stackrel{\circ }{\gamma }_{AB}=\frac{\delta _{AB}}{\left(1+\frac{1}{4}|\theta |^2\right)^2}.$$

Clearly, it suffices to construct initial data on $H_0$ (with $0\leq \underline{u}< \underline{u}_*$ for $\underline{u}_*\leq \epsilon$). The construction on $\underline{H}_0$ is similar. On $H_0$, we set $\Omega =1$ and therefore $e_4=\frac{\partial }{\partial \underline{u}}$. We will construct a metric on $H_0$ in the $(\underline{u},\theta ^1,\theta ^2)$ coordinates taking the form

$$\begin{eqnarray} {\gamma }_{AB}=\Phi ^2\hat{\gamma }_{AB},{\quad \text{where }\hat{\gamma }_{AB}=\frac{m_{AB}}{\left(1+\frac{1}{4}|\theta |^2\right)^2}}, {\tag{19}} \end{eqnarray}$$

and $\det m_{AB} =1$ and $\Phi \restriction _{S_{0,0}}=1$. In order to ensure that $m$ satisfies $\det m=1$, we write

$$m=\exp \Psi ,$$

with $\Psi \in \hat{S}$, where $\hat{S}$ denotes the set of all matrices taking the form

$$\begin{equation*} \left( \begin{array}{cc} a & b \\b & -a \end{array} \right). \end{equation*}$$

We will impose upper and lower bounds on $\Psi$. Since there are no smooth globally non-vanishing $\Psi \in \hat{S}$ on the 2-sphere, we use the convention that $\lesssim$ denotes that the quantity is bounded above by a uniform constant, while $\sim$ denotes that the quantity is bounded above by a uniform constant, and is bounded below at every $(\theta ^1, \theta ^2)$ by a constant depending on $(\theta ^1, \theta ^2)$ (where the constant is moreover allowed to vanish at finitely many isolated points). We require $\Psi \in \hat{S}$ to satisfy⁠²²

²²

Here and in the rest of this section, we use the notation that $J=(j_1,j_2)\in (\mathbb{N}\cup \{0\})\times (\mathbb{N}\cup \{0\})$ is a multi-index and $(\frac{\partial }{\partial \theta })^J=(\frac{\partial }{\partial \theta ^1})^{j_1}(\frac{\partial }{\partial \theta ^2})^{j_2}$. We moreover denote $|J|=j_1+j_2$.

✖

$$\begin{equation} {\sum _{|J|\leq N} \left|\left(\frac{\partial }{\partial \theta }\right)^J\Psi \right| \lesssim 1,}\qquad \sum _{|J|\leq N} \left|\left(\frac{\partial }{\partial \theta }\right)^J\frac{\partial }{\partial \underline{u}} \Psi \right| \lesssim f(\underline{u})^{-2}{,\quad \left|\frac{\partial }{\partial \underline{u}} \Psi \right| \sim f(\underline{u})^{-2}} {\tag{20}} \end{equation}$$

for some sufficiently large integer $N$. Following 5, we have

$$\begin{eqnarray} \hat{\chi }_{AB}=\frac{1}{2} \Phi ^2 \frac{\partial }{\partial \underline{u}}\hat{\gamma }_{AB},\quad \mathrm{tr\,}\chi =\frac{2}{\Phi }\frac{\partial \Phi }{\partial \underline{u}}. {\tag{21}} \end{eqnarray}$$

We can also derive that

$$\left\|\hat{\chi }\right\|_\gamma ^2=\frac{1}{4} \left({\hat{\gamma }}^{-1}\right)^{AC}\left({\hat{\gamma }}^{-1}\right)^{BD}\frac{\partial }{\partial \underline{u}}\hat{\gamma }_{AB}\frac{\partial }{\partial \underline{u}}\hat{\gamma }_{CD}.$$

Thus by 20, we have

$$\begin{eqnarray} |\hat{\chi }|_\gamma ^2 \sim f(\underline{u})^{-4}. {\tag{22}} \end{eqnarray}$$

In particular this implies the requirement in Theorem 4 is satisfied if $\int _0^{\underline{u}_*} f(\underline{u})^{{-}4} d\underline{u}= \infty$. By the equation

$${\mathcal{L}} \mkern -9mu /\,_{\frac{\partial }{\partial \underline{u}}}\mathrm{tr\,}\chi =-\frac{1}{2} (\mathrm{tr\,}\chi )^2 -|\hat{\chi }|^2,$$

$\Phi$ can be solved from the ODE

$$\begin{eqnarray} \frac{\partial ^2\Phi }{\partial \underline{u}^2}+\frac{1}{8} \left(\left({\hat{\gamma }}^{-1}\right)^{AC}\left({\hat{\gamma }}^{-1}\right)^{BD}\frac{\partial }{\partial \underline{u}}\hat{\gamma }_{AB}\frac{\partial }{\partial \underline{u}}\hat{\gamma }_{CD}\right)\Phi =0. {\tag{23}} \end{eqnarray}$$

We prescribe $\mathrm{tr\,}\chi$ on $S_{0,0}$ to obey the initial conditions

$$\begin{eqnarray} {\Phi \restriction _{S_{0,0}}=1,\qquad \frac{\partial \Phi }{\partial \underline{u}}\restriction _{S_{0,0}}=\frac{1}{2} \mathrm{tr\,}\chi \restriction _{S_{0,0}}\lesssim 1.} {\tag{24}} \end{eqnarray}$$

Finally, we prescribe $\zeta$ on $S_{0,0}$ such that

$$\begin{eqnarray} \sum _{|J|\leq N-1}\left|\left(\frac{\partial }{\partial \theta }\right)^J\zeta \right|_\gamma ^2 \lesssim 1. {\tag{25}} \end{eqnarray}$$

We check that these initial data obey all the estimates required by Theorem 2:

Estimates for $\nabla ^i\chi$ and the metric

To satisfy the upper bounds in Theorem 2, we need to show that

$$\begin{eqnarray} \sum _{i\leq N}\left|\nabla ^i\chi \right|_{\gamma }(0,\underline{u})\lesssim f(\underline{u})^{-2}. {\tag{26}} \end{eqnarray}$$

We will show the estimates separately for $\mathrm{tr\,}\chi$ and $\hat{\chi }$. By 22, 26 holds for $\hat{\chi }$ when $i=0$. To derive this bound for $\mathrm{tr\,}\chi$, notice that by the ODE 23 for $\Phi$, the initial conditions 24, and the bound 22 for $|\hat{\chi }|^2$, we have

$$\begin{eqnarray} \frac{1}{2}\leq \Phi \leq 1 {\tag{27}} \end{eqnarray}$$

and

$$\left|\frac{\partial \Phi }{\partial \underline{u}}\right|\lesssim 1+\int _0^{\underline{u}} f\left(\underline{u}'\right)^{-4} d\underline{u}'\leq 1+f\left(\underline{u}\right)^{-2}\int _0^{\underline{u}_*} f\left(\underline{u}'\right)^{-2} d\underline{u}'\leq 1+\epsilon ^2 f\left(\underline{u}\right)^{-2}$$

for $\epsilon$ sufficiently small. In the above estimate, we have used $\int _0^{\underline{u}_*} f(\underline{u}')^{-2}\, d\underline{u}'\leq \epsilon ^2$. By 21, we thus have

$$\left\|\mathrm{tr\,}\chi \right\|\lesssim f(\underline{u})^{-2}.$$

We now move on to control the angular derivatives of $\chi$. By 20,

$$\sum _{|J|\leq N}\left| \left(\frac{\partial }{\partial \theta }\right)^J\frac{\partial }{\partial \underline{u}} m_{AB} \right| \lesssim f\left(\underline{u}\right)^{-2}.$$

Using this bound and commuting the ODE 23 with $\frac{\partial }{\partial \theta }$, we also have that for up to $N$ coordinate angular derivatives $\frac{\partial }{\partial \theta }$,

$$\begin{eqnarray} \sum _{|J|\leq N}\left|\left(\frac{\partial }{\partial \theta }\right)^J\Phi \right| {\lesssim } 1. {\tag{28}} \end{eqnarray}$$

This implies via 19 and 20 that the metric $\gamma$ obeys the bounds

$$\begin{eqnarray} \displaystyle \sum _{|J|\leq N}\left|\left(\frac{\partial }{\partial \theta }\right)^J\gamma _{AB}\right| {\lesssim } 1,\qquad \sum _{|J|\leq N}\left|\left(\frac{\partial }{\partial \theta }\right)^J(\gamma ^{-1})^{AB}\right| {\lesssim } 1. {\tag{29}} \end{eqnarray}$$

Together with 20 and 21, 28 implies

$$\begin{eqnarray} \sum _{|J|\leq N}\left|\left(\frac{\partial }{\partial \theta }\right)^J\hat{\chi }\right| \lesssim f(\underline{u})^{-2}. {\tag{30}} \end{eqnarray}$$

By 21, we also have

$$\begin{eqnarray} \sum _{|J|\leq N}\left|\left(\frac{\partial }{\partial \theta }\right)^J\mathrm{tr\,}\chi \right|\lesssim f(\underline{u})^{-2}. {\tag{31}} \end{eqnarray}$$

Finally, we notice that by 29, the angular covariant derivatives of $\mathrm{tr\,}\chi$ and $\hat{\chi }$ can be controlled by the angular coordinate derivatives of $\mathrm{tr\,}\chi$ and $\hat{\chi }$. Therefore, 26 follows from 30 and 31.

Estimates for $\nabla ^i K$

To control $\nabla ^i K$, we simply notice that by 29, we have

$$\sum _{i\leq N-2}\left|\nabla ^i K\right|_{\gamma }\lesssim 1.$$

Estimates for $\nabla ^i\zeta$

On $H_0$, since $\Omega =1$, $\eta =\zeta$. Thus combining the transport equation for $\zeta$ in 9 and the Codazzi equation for $\beta$ in 10, and rewriting in ${\mathcal{L}} \mkern -9mu /\,$ (instead of $\nabla _4$), we have

$${\mathcal{L}} \mkern -9mu /\,_{\frac{\partial }{\partial \underline{u}}} \zeta +\mathrm{tr\,}\chi \zeta =\mathrm{div\,}\chi -\nabla \mathrm{tr\,}\chi .$$

Recall from 25 that the initial data for $\zeta$ and its angular derivatives are bounded. Therefore, by the estimates for $\mathrm{tr\,}\chi$ and $\hat{\chi }$ (and their angular derivatives) above, we have

$$\sum _{|J|\leq N-1}\left\|\left(\frac{\partial }{\partial \theta }\right)^J\zeta \right\|\lesssim 1.$$

The bounds for the metric and Christoffel symbols on the sphere imply

$$\sum _{j\leq N-1}\left\|\nabla ^j\zeta \right\|_{L^\infty _{\underline{u}}L^\infty (S_{0,\underline{u}})}\lesssim 1$$

as desired.

Estimates for $\nabla ^i \mathrm{tr\,}\underline{\chi }$

Similarly to $\zeta$, $\mathrm{tr\,}\underline{\chi }$ obeys a transport equations along the null generators of $H_0$. More precisely, 9 and the Gauss equation in 10 imply that

$${\mathcal{L}} \mkern -9mu /\,_{\frac{\partial }{\partial \underline{u}}} \mathrm{tr\,}\underline{\chi }+\mathrm{tr\,}\chi \mathrm{tr\,}\underline{\chi }=-2K-2\mathrm{div\,}\zeta +2|\zeta |^2.$$

Thus, the previous estimates imply

$$\sum _{j\leq N-2}\left\|\nabla ^j\mathrm{tr\,}\underline{\chi }\right\|_{L^\infty _{\underline{u}}L^\infty (S_{0,\underline{u}})}\lesssim 1.$$

Now, combining all the estimates that we have obtained so far, requiring $f$ to satisfy

$$\int _0^{\underline{u}_*} f(\underline{u})^{-4} d\underline{u}=\infty$$

and taking $N$ to sufficiently large, we have thus constructed initial data set on $H_0$ that obeys the assumptions of Theorems 2 and 4 on $H_0$. As mentioned above, it is easy to construct initial data set analogously on $\underline{H}_0$ so that the full set of assumptions of Theorems 2 and 4 are satisfied.

5. The preliminary estimates

We now turn to the proof Theorem 5, which will form the content of sections 5–8. In this section we derive the necessary preliminary estimates. In section 6 (see Proposition 15), we will prove the bound

$$\sum _{i\leq 3}\mathcal{O}_{i,2}\leq C(\mathcal{O}_{\mathrm{ini}} );$$

in section 7 (see Proposition 25), we will prove

$$\tilde{\mathcal{O}}_{4,2}\leq C(\mathcal{O}_{\mathrm{ini}} )(1+\mathcal{R});$$

and in section 8 (see Proposition 32), we will derive the estimate

$$\mathcal{R}\leq C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ).$$

Combining these estimates then implies the conclusion of Theorem 5.

We now begin with the preliminary estimates. All estimates in this section will be proved under the bootstrap assumption

$$\begin{equation} \sum _{i\leq 1}\mathcal{O}_{i,\infty }+\sum _{i\leq 2}\mathcal{O}_{i,4}+\sum _{i\leq 3}\mathcal{O}_{i,2}\leq \Delta _1, {\tag{A1}} \end{equation}$$

where $\Delta _1$ is a constant that will be chosen later.

5.1. Estimates for metric components

We first show that we can control $\Omega$ under the bootstrap assumption (A1):

Proposition 1.

There exists $\epsilon _0=\epsilon _0(\Delta _1)$ such that for every $\epsilon \leq \epsilon _0$,

$$\frac{1}{2}\leq \Omega \leq 2.$$

Moreover, $\Omega$ is continuous up to $u=u_*$ and $\underline{u}=\underline{u}_*$.

Proof.

Consider the equation

$$\begin{equation} \omega =-\frac{1}{2}\nabla _4\log \Omega =\frac{1}{2}\Omega \nabla _4\Omega ^{-1}=\frac{1}{2}\frac{\partial }{\partial \underline{u}}\Omega ^{-1}. {\tag{32}} \end{equation}$$

Fix $\underline{u}$. Notice that both $\omega$ and $\Omega$ are scalars and therefore the $L^\infty$ norm is independent of the metric. We can integrate equation (32) using the fact that $\Omega ^{-1}=1$ on $\underline{H}_0$ to obtain

$$\begin{equation*} \begin{split} &\left\|\Omega ^{-1}-1\right\|_{L^\infty (S_{u,\underline{u}})}\\ &\qquad \leq C\int _0^{\underline{u}}\left\|\omega \right\|_{L^\infty \left(S_{u,\underline{u}'}\right)}d\underline{u}'\leq C\left\|f(\underline{u})^{-1}\right\|_{L^2_{\underline{u}}}\left\|f(\underline{u})\omega \right\|_{L^\infty _uL^2_{\underline{u}}L^\infty (S)}\leq C\Delta _1\epsilon . \end{split} \end{equation*}$$

This implies both the upper and lower bounds for $\Omega$ for sufficiently small $\epsilon$. To show continuity, take a sequence of points $(u_n, \underline{u}_n, \theta ^1_n, \theta ^2_n)$ such $u_n \to u_\infty$, $\underline{u}_n\to \underline{u}_\infty$, $\theta _n^1\to \theta _\infty ^1$, and $\theta _n^2\to \theta _\infty ^2$. Then

$$\begin{equation*} \begin{split} &\left|\Omega ^{-1}\left(u_n,\underline{u}_n,\theta _n^1,\theta _n^2\right)-\Omega ^{-1}\left(u_m,\underline{u}_m,\theta _m^1,\theta _m^2\right)\right|\\ &\qquad \leq \left|\Omega ^{-1}\left(u_n,\underline{u}_n,\theta _n^1,\theta _n^2\right)-\Omega ^{-1}\left(u_n,\underline{u}_n,\theta _m^1,\theta _m^2\right)\right|\\ &\qquad \quad +\left|\Omega ^{-1}\left(u_n,\underline{u}_n,\theta _m^1,\theta _m^2\right)-\Omega ^{-1}\left(u_n,\underline{u}_m,\theta _m^1,\theta _m^2\right)\right|\\ &\qquad \quad +\left|\Omega ^{-1}\left(u_n,\underline{u}_m,\theta _m^1,\theta _m^2\right)-\Omega ^{-1}\left(u_m,\underline{u}_m,\theta _m^1,\theta _m^2\right)\right|\\ &\qquad \leq C\left\|\nabla \log \Omega \right\|_{L^\infty \left(S_{u_n,\underline{u}_n}\right)}\text{dist}_{S_{u_n,\underline{u}_n}}\left(\theta _n,\theta _m\right)+2\left|\int _{\underline{u}_n}^{\underline{u}_m} \left\|\omega \right\|_{L^\infty \left(S_{u_n,\underline{u}'}\right)} d\underline{u}'\right|\\ &\qquad \quad +2\left|\int _{u_n}^{u_m} \left\|\underline{\omega }\right\|_{L^\infty \left(S_{u',\underline{u}_m}\right)} du'\right|. \end{split} \end{equation*}$$

Since by the bootstrap assumption A1, $\nabla \log \Omega =\frac{1}{2}(\eta +\underline{\eta })$ is uniformly bounded, $||\omega ||_{L^\infty (S_{u,\underline{u}})}$ is uniformly integrable in $\underline{u}$ for all $u,$ and $||\underline{\omega }||_{L^\infty (S_{u,\underline{u}})}$ is uniformly integrable in $u$ for all $\underline{u}$, the right-hand side can be made arbitrarily small by taking $n,m \geq N$ for $N$ sufficiently large. The conclusion thus follows.

■

We then show that we can control $\gamma$ under the bootstrap assumption (A1):

Proposition 2.

There exists $\epsilon _0=\epsilon _0(\Delta _1)$ such that for $\epsilon \leq \epsilon _0$, in the $(u,\underline{u},\theta ^1,\theta ^2)$ coordinate system, we have

$$c\leq \det \gamma \leq C,\quad \left|\gamma _{AB}\right|,\left|\left(\gamma ^{-1}\right)^{AB}\right|\leq C{,}$$

where the constants depend only on $d$ and $D$. Moreover, $\gamma$ remains continuous up to $u=u_*$ and $\underline{u}=\underline{u}_*$.

Proof.

We first prove the bound for $\gamma$ on the initial hypersurface $\underline{H}_0$. Using

$${\mathcal{L}} \mkern -9mu /\,_{\underline{L}}\gamma =2\Omega \underline{\chi },$$

we get⁠²³

²³

Note that $b^A=0$ on $\underline{H}_0$.

✖

$$\frac{\partial }{\partial u}\gamma _{AB}=2\Omega \underline{\chi }_{AB},\qquad \frac{\partial }{\partial u}\log (\det \gamma )=\Omega \mathrm{tr\,}\underline{\chi }$$

on $\underline{H}_0$. We therefore have

$$\begin{equation} \left|\frac{\det \gamma (u,0)}{\det \gamma (0,0)}\right|\leq C\exp \left(\int _0^{u}\left|\mathrm{tr\,}\underline{\chi }\right|(u',0)\,du'\right)\leq C(D). {\tag{33}} \end{equation}$$

This implies that the $\det \gamma$ is bounded above and below. Let $\Lambda$ be the larger eigenvalue of $\gamma$. Clearly,

$$\begin{equation} \Lambda \leq C\sup _{A,B=1,2}\left|\gamma _{AB}\right|,\qquad \sum _{A,B=1,2}\left|\underline{\chi }_{AB}\right|^2\leq C\Lambda ^2 \left\|\underline{\chi }\right\|_{L^\infty \left(S_{u,\underline{u}}\right)}^2. {\tag{34}} \end{equation}$$

Then

$$\begin{equation} \begin{split} &|\gamma _{AB}(u,0)-(\gamma )_{AB}(0,0)|\\ &\qquad \leq C\epsilon \left(\sup _{u'\leq u}\Lambda \right)\left(\int _0^{u}f(u')^2\left\|\underline{\chi }\right\|_{L^\infty \left(S_{u',0}\right)}^2du'\right)^{\frac{1}{2}}\leq C(D)\left(\sup _{u'\leq u}\Lambda \right)\epsilon . \end{split} {\tag{35}} \end{equation}$$

Using the first upper bound in (34), we thus obtain the upper bound for $|\gamma _{AB}|$ after choosing $\epsilon$ to be sufficiently small. The upper bound for $|(\gamma ^{-1})^{AB}|$ follows from the upper bound for $|\gamma _{AB}|$ and the lower bound for $\det \gamma$.

Now, in order to obtain the bounds for $\gamma _{AB}$ in the spacetime, we argue similarly but use the propagation equation in the $\underline{u}$ direction and compare $\gamma (u,\underline{u})$ with $\gamma (u,0)$. Here, we use bootstrap assumption A1 instead of the assumptions on the initial data. More precisely, we have

$$\begin{equation} \frac{\partial }{\partial \underline{u}}\gamma _{AB}=2\Omega \chi _{AB},\qquad \frac{\partial }{\partial \underline{u}}\log (\det \gamma )=\Omega \mathrm{tr\,}\chi . {\tag{36}} \end{equation}$$

We then derive as above that

$$\begin{equation*} \left|\frac{\det \gamma (u,\underline{u})}{\det \gamma (u,0)}\right|\leq C\exp (C\Delta _1\epsilon ),\quad \left|\gamma _{AB}(u,\underline{u})-\gamma _{AB}(u,0)\right|\leq C\left(\sup _{\substack{u'\leq u\\ \underline{u}'\leq \underline{u}}}\Lambda \right)\Delta _1\epsilon , \end{equation*}$$

where $\Lambda$ is the larger eigenvalue for $\gamma _{AB}$. As before, we thus obtain the upper bounds for $|\gamma _{AB}|$ and $|(\gamma ^{-1})^{AB}|$. Finally, the continuity of $\gamma$ up to the boundary follows as in the proof of continuity for $\Omega$ in Proposition 1.

■

With the estimates on $\gamma$, it follows that the $L^p$ norms defined with respect to the metric and the $L^p$ norms defined with respect to the coordinate system are equivalent.

Proposition 3.

Given a covariant tensor $\phi _{A_1\cdots A_r}$ on $S_{u,\underline{u}}$, we have

$$\int _{S_{u,\underline{u}}} \langle \phi ,\phi \rangle _{\gamma }^{p/2} \sim \sum _{A_i=1,2}\iint \left|\phi _{A_1\cdots A_r}\right|^p \sqrt {\det \gamma } d\theta ^1 d\theta ^2.$$

We can also bound $b$ under the bootstrap assumption, thus controlling the full spacetime metric:

Proposition 4.

In the coordinate system $(u,\underline{u},\theta ^1,\theta ^2)$,

$$\left|b^A\right|\leq C\Delta _1\epsilon .$$

Moreover, $b^A$ is continuous up to $u=u_*$ and $\underline{u}=\underline{u}_*$.

Proof.

$b^A$ satisfies the equation

$$\begin{equation} \frac{\partial b^A}{\partial \underline{u}}=-4\Omega ^2\zeta ^A. {\tag{37}} \end{equation}$$

This can be derived from

$$\left[L,\underline{L}\right]=\frac{\partial b^A}{\partial \underline{u}}\frac{\partial }{\partial \theta ^A}.$$

Now, integrating 37 and using Proposition 3 gives the bound on $b$. Continuity of $b$ up to the boundary follows as in the proof of Proposition 1.

■

5.2. Estimates for transport equations

In this subsection, we prove general propositions for obtaining bounds from the covariant null transport equations. Such estimates require the integrability of $\mathrm{tr\,}\chi$ and $\mathrm{tr\,}\underline{\chi }$, which is consistent with our bootstrap assumption (A1). This will be used in the following sections to derive some estimates for the Ricci coefficients and the null curvature components from the null structure equations and the null Bianchi equations, respectively. Below, we state two propositions which provide $L^p$ estimates for general quantities satisfying transport equations either in the $e_3$ or $e_4$ direction.

Proposition 5.

There exists $\epsilon _0=\epsilon _0(\Delta _1)$ such that for all $\epsilon \leq \epsilon _0$ and for every $2\leq p<\infty$, we have

$$\begin{equation*} ||\phi ||_{L^p(S_{u,\underline{u}})}\leq C(||\phi ||_{L^p(S_{u,\underline{u}'})}+\int _{\underline{u}'}^{\underline{u}} ||\nabla _4\phi ||_{L^p(S_{u,\underline{u}''})}d{\underline{u}''}), \end{equation*}$$

$$\begin{equation*} ||\phi ||_{L^p(S_{u,\underline{u}})}\leq C(||\phi ||_{L^p(S_{u',\underline{u}})}+\int _{u'}^{u} ||\nabla _3\phi ||_{L^p(S_{u'',\underline{u}})}d{u''}) \end{equation*}$$ for any tensor $\phi$ tangential to the spheres $S_{u,\underline{u}}$.

Proof.

The following identity⁠²⁴ holds for any scalar $f$:

²⁴

Here, $\frac{\partial }{\partial \underline{u}}$ on the left-hand side is to be understood as the coordinate vector field in the $(u,\underline{u})$-plane. Similarly for $\frac{\partial }{\partial u}$ below.

✖

$$\begin{equation*} \frac{\partial }{\partial \underline{u}}\int _{\mathcal{S}_{u,\underline{u}}} f=\int _{\mathcal{S}_{u,\underline{u}}} \Omega \left(e_4(f)+ \mathrm{tr\,}\chi f\right). \end{equation*}$$

Similarly, we have

$$\begin{equation*} \frac{\partial }{\partial u}\int _{\mathcal{S}_{u,\underline{u}}} f=\int _{\mathcal{S}_{u,\underline{u}}} \Omega \left(e_3(f)+ \mathrm{tr\,}\underline{\chi }f\right). \end{equation*}$$

Hence, taking $f=|\phi |_{\gamma }^p$, we have

$$\begin{equation} \begin{split} ||\phi ||^p_{L^p(S_{u,\underline{u}})}=&\ ||\phi ||^p_{L^p(S_{u,\underline{u}'})}+\int _{\underline{u}'}^{\underline{u}}\int _{S_{u,\underline{u}''}} p|\phi |^{p-2}\Omega \left(\langle \phi ,\nabla _4\phi \rangle _\gamma + \frac{1}{p}\mathrm{tr\,}\chi |\phi |^2_{\gamma }\right)d{\underline{u}''}{,}\\ ||\phi ||^p_{L^p(S_{u,\underline{u}})}=&\ ||\phi ||^p_{L^p(S_{u',\underline{u}})}+\int _{u'}^{u}\int _{S_{u'',\underline{u}}} p|\phi |^{p-2}\Omega \left(\langle \phi ,\nabla _3\phi \rangle _\gamma + \frac{1}{p}\mathrm{tr\,}\underline{\chi }|\phi |^2_{\gamma }\right)d{u''}{.} \end{split} {\tag{38}} \end{equation}$$

The bootstrap assumption (A1) implies that $\mathrm{tr\,}\chi$ and $\mathrm{tr\,}\underline{\chi }$ are integrable (and in fact it also implies that $||\mathrm{tr\,}\chi ||_{L^1_{\underline{u}}L^\infty _u L^\infty (S)}$ and $||\mathrm{tr\,}\underline{\chi }||_{L^1_{u}L^\infty _{\underline{u}} L^\infty (S)}$ are small after choosing $\epsilon$ to be small depending on $\Delta _1$). Thus the proposition can be proved by using Hölder’s inequality and Gronwall’s inequality, together with the bound for $\Omega$ given in Proposition 1.

■

We also have the following bounds for the $p=\infty$ case by integrating along the integral curves of $e_3$ and $e_4$:

Proposition 6.

There exists $\epsilon _0=\epsilon _0(\Delta _1)$ such that for all $\epsilon \leq \epsilon _0$, we have

$$\begin{equation*} ||\phi ||_{L^\infty (S_{u,\underline{u}})}\leq C\left(||\phi ||_{L^\infty (S_{u,\underline{u}'})}+\int _{\underline{u}'}^{\underline{u}} ||\nabla _4\phi ||_{L^\infty (S_{u,\underline{u}''})}d{\underline{u}''}\right), \end{equation*}$$

$$\begin{equation*} ||\phi ||_{L^\infty (S_{u,\underline{u}})}\leq C\left(||\phi ||_{L^\infty (S_{u',\underline{u}})}+\int _{u'}^{u} ||\nabla _3\phi ||_{L^\infty (S_{u'',\underline{u}})}d{u''}\right) \end{equation*}$$ for any tensor $\phi$ tangential to the spheres $S_{u,\underline{u}}$.

Proof.

This follows simply from integrating along the integral curves of $L$ and $\underline{L}$, and the estimate on $\Omega$ in Proposition 1.

■

5.3. Sobolev embedding

Using the estimates for the metric $\gamma$ in Proposition 2, we have the following Sobolev embedding theorem:

Proposition 7.

There exists $\epsilon _0=\epsilon _0(\Delta _1)$ such that as long as $\epsilon \leq \epsilon _0$, we have

$$||\phi ||_{L^4(S_{u,\underline{u}})}\leq C\sum _{i=0}^1||\nabla ^i\phi ||_{L^2(S_{u,\underline{u}})}$$

and

$$||\phi ||_{L^\infty (S_{u,\underline{u}})}\leq C\left(||\phi ||_{L^2(S_{u,\underline{u}})}+||\nabla \phi ||_{L^4(S_{u,\underline{u}})}\right)$$

for any tensor $\phi$ tangential to the spheres $S_{u,\underline{u}}$. Combining the above estimates, we also have

$$||\phi ||_{L^\infty (S_{u,\underline{u}})}\leq C\sum _{i=0}^2||\nabla ^i\phi ||_{L^2(S_{u,\underline{u}})}.$$

Proof.

By 35 in the proof of Proposition 2, $|\gamma _{AB}(u,\underline{u})-\gamma _{AB}(0,0)|$ can be made arbitrarily small by choosing $\epsilon$ to be small. Therefore, the isoperimetric constant

$$I(S_{u,\underline{u}})=\sup _U\frac{\min \{\text{Area}(U),\text{Area}(U^c)\}}{\text{Perimeter}(\partial U)}$$

on every sphere $S_{u,\underline{u}}$ is controlled⁠²⁵ up to a constant factor by the corresponding isoperimetric constant on $S_{0,0}$. Once the isoperimetric constants are uniformly controlled, the Sobolev embedding theorem follows from 5, Lemmas 5.1 and 5.2 and the fact that the volume of $S_{u,\underline{u}}$ is bounded uniformly above and below.

²⁵

This argument is standard. We refer the readers for instance to 5, Lemma 5.4.

✖

■

5.4. Commutation formulae

We have the following formula from 6, Lemma 7.3.3:

Proposition 8.

The commutator $[\nabla _4,\nabla ]$ acting on a rank $r$ tensor $\phi$ tangential to the spheres $S_{u,\underline{u}}$ is given by

$$\begin{equation*} \begin{split} &[\nabla _4,\nabla _B]\phi _{A_1\cdots A_r}\\ &\qquad =(\nabla _B\log \Omega )\nabla _4\phi _{A_1\cdots A_r}-(\gamma ^{-1})^{CD}\chi _{BD}\nabla _C\phi _{A_1\cdots A_r} \\ &\qquad \quad +\sum _{i=1}^r ((\gamma ^{-1})^{CD}\chi _{A_iB}\underline{\eta }_{D}-(\gamma ^{-1})^{CD}\chi _{BD}\underline{\eta }_{A_i}+{\epsilon } \mkern -7mu /\,_{A_i}{ }^C{ }^*\beta _B)\phi _{A_1\cdots \hat{A_i}C\cdots A_r}. \end{split} \end{equation*}$$

Similarly, the commutator $[\nabla _3,\nabla ]$ is given by

$$\begin{equation*} \begin{split} &[\nabla _3,\nabla _B]\phi _{A_1\cdots A_r}\\ &\qquad =(\nabla _B\log \Omega )\nabla _3\phi _{A_1\cdots A_r}-(\gamma ^{-1})^{CD}\underline{\chi }_{BD}\nabla _C\phi _{A_1\cdots A_r} \\ &\qquad \quad +\sum _{i=1}^r ((\gamma ^{-1})^{CD}\underline{\chi }_{A_iB}\eta _{D}-(\gamma ^{-1})^{CD}\underline{\chi }_{BD}\eta _{A_i}-{\epsilon } \mkern -7mu /\,_{A_i}{ }^C{ }^*\underline{\beta }_B)\phi _{A_1\cdots \hat{A_i}C\cdots A_r}. \end{split} \end{equation*}$$

Recall the schematic notation

$$\psi \in \{\eta ,\underline{\eta }\},\qquad \psi _H\in \{\mathrm{tr\,}\chi ,\hat{\chi },\omega \},\qquad \psi _{\underline{H}}\in \{\mathrm{tr\,}\underline{\chi },\hat{\underline{\chi }},\underline{\omega }\}.$$

By induction and the schematic Codazzi equations

$$\beta =\nabla \chi +\psi \chi =\nabla \psi _H+\psi \psi _H,\qquad {\underline{\beta }}=\nabla \underline{\chi }+\psi \underline{\chi }=\nabla \psi _{\underline{H}}+\psi \psi _{\underline{H}},{}$$

we get the following schematic formula for repeated commutations (see 19):

Proposition 9.

Suppose $\nabla _4\phi =F_0$ for some tensors $\phi$ and $F_0$. Let $F_i$ be the tensor defined by $\nabla _4\nabla ^i\phi =F_i$. Then

$$\begin{equation*} \begin{split} F_i\sim &\sum _{i_1+i_2+i_3=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3} F_0+\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4} \phi . \end{split} \end{equation*}$$

Similarly, suppose $\nabla _3\phi =G_{0}$ for some tensors $\phi$ and $G_0$. Let $G_i$ be the tensor defined by $\nabla _3\nabla ^i\phi =G_{i}$. Then

$$\begin{equation*} \begin{split} G_{i}\sim &\sum _{i_1+i_2+i_3=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3} G_{0}+\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4} \phi . \end{split} \end{equation*}$$

5.5. General elliptic estimates for Hodge systems

We recall the definition of the divergence and curl of a symmetric covariant tensor of rank $r+1$:

$$(\mathrm{div\,}\phi )_{A_1\cdots A_r}=\nabla ^B\phi _{BA_1\cdots A_r},$$

$$(\mathrm{curl\,}\phi )_{A_1\cdots A_r}={\epsilon } \mkern -7mu /\,^{BC}\nabla _B\phi _{CA_1\cdots A_r},$$ where ${\epsilon } \mkern -7mu /\,$ is the volume form associated to the metric $\gamma$. Recall also that the trace is defined to be

$$(tr\phi )_{A_1\cdots A_{r-1}}=(\gamma ^{-1})^{BC}\phi _{BCA_1\cdots A_{r-1}}.$$

The following elliptic estimate is standard (see, for example, 6, Lemmas 2.2.2, 2.2.3 or 5, Lemmas 7.1, 7.2, 7.3):

Proposition 10.

Let $\phi$ be a symmetric $r$ covariant tensor on a 2-sphere $(\mathbb{S}^2,\gamma )$ satisfying

$$\mathrm{div\,}\phi =f,\qquad \mathrm{curl\,}\phi =g,\qquad tr\phi =h.$$

Suppose also that

$$\sum _{i\leq 2}||\nabla ^i K||_{L^2(S)}< \infty .$$

Then for $i\leq 4$, there exists a constant $C_E$ depending only on $\sum _{i\leq 2}||\nabla ^i K||_{L^2(S)}$ such that

$$||\nabla ^{i}\phi ||_{L^2(S)}\leq C_E\left(\sum _{j=0}^{i-1}(||\nabla ^{j}f||_{L^2(S)}+||\nabla ^{j}g||_{L^2(S)}+||\nabla ^{j}h||_{L^2(S)}+||{\nabla ^j}\phi ||_{L^2(S)})\right)\!\!.$$

For the special case that $\phi$ is a symmetric traceless 2-tensor, we only need to know its divergence:

Proposition 11.

Suppose $\phi$ is a symmetric traceless 2-tensor satisfying

$$\mathrm{div\,}\phi =f.$$

Suppose moreover that

$$\sum _{i\leq 2}||\nabla ^i K||_{L^2(S)}< \infty .$$

Then for $i\leq 4$, there exists a constant $C_E$ depending only on $\sum _{i\leq 2}||\nabla ^i K||_{L^2(S)}$ such that

$$||\nabla ^{i}\phi ||_{L^2(S)}\leq C_E\left(\sum _{j=0}^{i-1}(||\nabla ^{j}f||_{L^2(S)}+||{\nabla ^j}\phi ||_{L^2(S)})\right).$$

Proof.

This follows from Proposition 10 and the fact that

$$\begin{equation*} \mathrm{curl\,}\phi =^*f. \end{equation*}$$■

6. Estimates for the Ricci coefficients via transport equations

In this section we prove $L^2$ estimates for the Ricci coefficients and their first, second, and third derivatives. We will assume bounds for $\mathcal{R}$ and $\tilde{\mathcal{O}}_{4,2}$ and show that for $\epsilon _0$ chosen to be sufficiently small, $\sum _{i\leq 3}\mathcal{O}_{i,2}$ is likewise bounded. In order to achieve this, we continue to work under the bootstrap assumption (A1) and will show that the constant in A1 can in fact be improved (see Proposition 15).

Recall that we will use the following notation: $\psi \in \{\eta ,\underline{\eta }\}$, $\psi _{\underline{H}}\in \{\mathrm{tr\,}\underline{\chi },\hat{\underline{\chi }},\underline{\omega }\},$ and $\psi _H\in \{\mathrm{tr\,}\chi ,\hat{\chi },\omega \}$.

We first show bounds for $\psi$.

Proposition 12.

Assume

$$\mathcal{R} <\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\Delta _1,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\begin{equation*} \sum _{i\leq 3}\mathcal{O}_{i,2}[\psi ]\leq C(\mathcal{O}_{\mathrm{ini}} ), \end{equation*}$$

i.e., the bounds depends only on the initial data norm $\mathcal{O}_{\mathrm{ini}}$. In particular $C(\mathcal{O}_{\mathrm{ini}} )$ is independent of $\Delta _1$.

Proof.

We first estimate $\eta$; the estimates for $\underline{\eta }$ are similar after we replace $u$ with $\underline{u}$ and $3$ with $4$. Using the null structure equations, we have a schematic equation of the type

$$\nabla _{4}\eta =\beta +\psi _H\psi .$$

We also commute the null structure equations with angular derivatives to get

$$\begin{equation} \nabla _{4}\nabla ^i\eta =\sum _{i_1+i_2+i_3=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\beta +\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _H. {\tag{39}} \end{equation}$$

By Proposition 5 in order to estimate $||\nabla ^i\eta ||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}$, it suffices to estimate the initial data and the $||\cdot ||_{L^\infty _uL^1_{\underline{u}}L^2(S)}$ norm of the right-hand side 39. Using the bootstrap assumption, we will show that the right-hand side is bounded in a weighted $L^2_{\underline{u}}$ norm. This in turn implies via an application of the Cauchy–Schwarz inequality that the $L^1_{\underline{u}}$ norm is also bounded. We now turn to the details.

We first estimate the curvature term

$$\sum _{i_1+i_2+i_3\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\beta .$$

For the terms such that at most $1$ derivative falling on $\psi$, the bootstrap assumption A1 allows us to control $\sum _{i\leq 1}\|\nabla ^i\psi \|_{L^\infty _uL^\infty _{\underline{u}}L^\infty (S)}$ by $\Delta _1$. We then need to control $\sum _{i\leq 3}\nabla ^i\beta$ in $L^\infty _uL^1_{\underline{u}}L^2(S)$. By the Cauchy–Schwarz inequality, since the $L^2_{\underline{u}}$ norm of $f(\underline{u})^{-1}$ is smaller than $\epsilon$, we can bound this by $\sum _{i\leq 3}\nabla ^i\beta$ in the weighted $L^2$ norms. More precisely, we have

$$\begin{equation} \begin{split} &\left\|\sum _{i_1\leq 1,i_2\leq 3,i_3\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\beta \right\|_{L^\infty _{u}L^1_{\underline{u}}L^2(S)} \\ &\quad \leq C\left(\sum _{i_1\leq 1,i_2\leq 3}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{u}L^\infty _{\underline{u}}L^\infty (S)}\right)\left(\sum _{i_3\leq 3}||f(\underline{u})\nabla ^{i_3}\beta ||_{L^\infty _{u}L^1_{\underline{u}}L^2(S)}\right)\\ &\quad \leq C\left(\sum _{i_1\leq 1,i_2\leq 3}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{u}L^\infty _{\underline{u}}L^\infty (S)}\right)\\ &\qquad \times \left(\sum _{i_3\leq 3}||f(\underline{u})\nabla ^{i_3}\beta ||_{L^\infty _{u}L^2_{\underline{u}}L^2(S)}\right)||f(\underline{u})^{-1}||_{L^\infty _{u}L^2_{\underline{u}}L^\infty (S)}\\ &\quad \leq C\epsilon (1+\Delta _1)^3\mathcal{R}{.} \end{split} {\tag{40}} \end{equation}$$

For the term where exactly two derivatives fall on $\psi$ (notice that this is the highest number of derivatives that can fall on $\psi$), we control $\nabla ^2\psi$ in $L^\infty _uL^\infty _{\underline{u}}L^2(S)$ by $\Delta _1$ (using A1). Thus we are left with $\beta$ in $L^\infty _uL^1_{\underline{u}}L^\infty (S)$. By Sobolev embedding (Proposition 7), this can be bounded by $\sum _{i\leq 3}\|\nabla ^i\beta \|_{L^\infty _u L^1_{\underline{u}}L^2(S)}$, which in turn can be controlled by $\mathcal{R}$ after applying the Cauchy–Schwarz inequality as in 40. More precisely,

$$\begin{equation} \begin{split} &||\nabla ^2\psi \beta ||_{L^\infty _{u}L^1_{\underline{u}}L^2(S)} \\ &\quad \leq C||\nabla ^2\psi ||_{L^\infty _{u}L^\infty _{\underline{u}}L^2(S)}||\beta ||_{L^\infty _{u}L^1_{\underline{u}}L^\infty (S)}\\ &\quad \leq C||\nabla ^2\psi ||_{L^\infty _{u}L^\infty _{\underline{u}}L^2(S)}\left(\sum _{i\leq 2}||\nabla ^i\beta ||_{L^\infty _{u}L^1_{\underline{u}}L^2(S)}\right)\\ &\quad \leq C||\nabla ^2\psi ||_{L^\infty _{u}L^\infty _{\underline{u}}L^2(S)}\left(\sum _{i\leq 2}||f(\underline{u})\nabla ^i\beta ||_{L^\infty _{u}L^2_{\underline{u}}L^\infty (S)}\right)||f(\underline{u})^{-1}||_{L^\infty _{u}L^2_{\underline{u}}L^\infty (S)}\\ &\quad \leq C\epsilon \Delta _1\mathcal{R}. \end{split} {\tag{41}} \end{equation}$$

Combining 40 and 41, we have

$$\left\|\sum _{i_1+i_2+i_3\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\beta \right\|_{L^\infty _{u}L^1_{\underline{u}}L^2(S)}\leq C\epsilon (1+\Delta _1)^3\mathcal{R}.$$

We then estimate the second term in 39. We separate the terms where more derivatives fall on $\psi _H$ and those where more derivatives fall on $\psi$:

$$\begin{equation} \begin{split} &\left\|\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _{H}\right\|_{L^\infty _{u}L^1_{\underline{u}}L^2(S)} \\ &\quad \leq C\left(\sum _{i_1\leq 1,{1\leq }i_2\leq 4}||\nabla ^{i_1}\psi ||_{L^\infty _{u}L^\infty _{\underline{u}}L^\infty (S)}^{i_2}\right) \left(\sum _{i_3\leq 3}||\nabla ^{i_3}\psi _H||_{L^\infty _{u}L^1_{\underline{u}}L^2(S)}\right) \\ &\quad \quad + C(1+||\psi ||_{L^\infty _u L^\infty _{\underline{u}}L^\infty (S)})\!\!\left(\sum _{i_1\leq 3}||\nabla ^{i_1}\psi ||_{L^\infty _{u}L^\infty _{\underline{u}}L^2(S)}\!\!\right)\!\!\left(\sum _{i_2\leq 1}||\nabla ^{i_2}\psi _H||_{L^\infty _{u}L^1_{\underline{u}}L^\infty (S)}\!\!\!\right)\\ &\quad \leq C\Delta _1(1+\Delta _1)^3\left(\sum _{{i}\leq 3}||\nabla ^{{i}}\psi _H||_{L^\infty _{u}L^1_{\underline{u}}L^2(S)}+\sum _{{i}\leq 1}||\nabla ^{{i}}\psi _H||_{L^\infty _{u}L^1_{\underline{u}}L^\infty (S)}\right)\\ &\quad \leq C\Delta _1(1+\Delta _1)^3||f(\underline{u})^{-1}||_{L^\infty _uL^2_{\underline{u}}L^\infty (S)}\\ &\qquad \qquad \times \left(\sum _{{i}\leq 3}||f(\underline{u})\nabla ^{{i}}\psi _H||_{L^\infty _{u}L^2_{\underline{u}}L^2(S)}+\sum _{{i}\leq 1}||f(\underline{u})\nabla ^{{i}}\psi _H||_{L^\infty _{u}L^2_{\underline{u}}L^\infty (S)}\right)\\ &\quad \leq C\Delta _1^2(1+\Delta _1)^3\epsilon . \end{split} {\tag{42}} \end{equation}$$

Hence, by Proposition 5, we have

$$\sum _{i\leq 3}||\nabla ^i\eta ||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )+C\epsilon (\Delta _1^2(1+\Delta _1)^3+\mathcal{R}(1+\Delta _1)^3)\leq C(\mathcal{O}_{\mathrm{ini}} ),$$

after choosing $\epsilon$ to be sufficiently small. Similarly, we consider the equation for $\nabla _3\nabla ^i\underline{\eta }$ to get

$$\begin{equation*} \sum _{i\leq 3}||\nabla ^i\underline{\eta }||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} ). \end{equation*}$$■

We now move to the terms that we denote by $\psi _{\underline{H}}$, i.e., $\mathrm{tr\,}\underline{\chi }$, $\hat{\underline{\chi }}$, and $\underline{\omega }$. All of them obey a $\nabla _4$ equation. Unlike the previous estimates for $\psi$, the initial data for the quantities $\psi _{\underline{H}}$ are not in $L^\infty _u$. We will therefore prove only a bound for $\psi _{\underline{H}}$ in the weighted norm $||f(u)\cdot ||_{L^2_u L^\infty _{\underline{u}}L^\infty (S)}$.

Proposition 13.

Assume

$$\mathcal{R} <\infty ,\quad \tilde{\mathcal{O}}_{4,2}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\Delta _1,\mathcal{R},\tilde{\mathcal{O}}_{4,2})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 3}\mathcal{O}_{i,2}[\psi _{\underline{H}}]\leq C(\mathcal{O}_{\mathrm{ini}} ) .$$

In particular as before, this estimate is independent of $\Delta _1$.

Proof.

According to the definition of the $\mathcal{O}_{i,2}$ norm, we need to control the weighted $L^2_{u}L^\infty _{\underline{u}}L^2(S)$ norm of $\psi _{\underline{H}}$. Using the null structure equations, for each $\psi _{\underline{H}}\in \{\mathrm{tr\,}\underline{\chi },\hat{\underline{\chi }},\underline{\omega }\}$, we have an equation of the type

$$\nabla _{4}\psi _{\underline{H}}=K+\nabla \underline{\eta }+\psi \psi +\psi _H\psi _{\underline{H}}.$$

We also use the null structure equations commuted with angular derivatives:

$$\begin{equation*} \begin{split} \nabla _{4}\nabla ^i\psi _{\underline{H}} &=\sum _{i_1+i_2+i_3=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}(K+\nabla \underline{\eta })+\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi \\ &\quad +\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}. \end{split} \end{equation*}$$

We estimate the curvature term using the curvature norm. Recall that the curvature norm for $K$ along the $H_u$ is weighted with $f(u)$. Using the Sobolev embedding theorem in Proposition 7, we have

$$\begin{equation} \begin{split} &\left\|\sum _{i_1+i_2+i_3\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}K\right\|_{L^1_{\underline{u}}L^2(S)} \\ &\qquad \leq C f(u)^{-1}\left(\sum _{i_1\leq 1,i_2\leq 3}||\nabla ^{i_1}\psi ||^{i_2}_{L^2_{\underline{u}}L^\infty (S)}\right)\left(\sum _{i_3\leq 3}||f(u)\nabla ^{i_3} K||_{L^2_{\underline{u}} L^2(S)}\right)\\ &\quad \qquad +C f(u)^{-1}||\nabla ^2\psi ||_{L^2_{\underline{u}}L^4(S)}||f(u) K||_{L^2_{\underline{u}} L^4(S)}\\ &\qquad \leq C {\epsilon ^{\frac{1}{2}}}f(u)^{-1}\left(\sum _{i_1\leq 3,i_2\leq 3}||\nabla ^{i_1}\psi ||^{i_2}_{L^{{\infty }}_{\underline{u}}L^2(S)}\right)\left(\sum _{i_3\leq 3}||f(u)\nabla ^{i_3} K||_{L^2_{\underline{u}} L^2(S)}\right)\\ &\qquad \leq C\epsilon ^{\frac{1}{2}} f(u)^{-1}(1+\Delta _1)^3\mathcal{R}. \end{split} {\tag{43}} \end{equation}$$

The term linear in $\nabla ^{4}\eta$ can be estimated analogously but using the $\tilde{\mathcal{O}}_{4,2}$ norms instead of the $\mathcal{R}$ norms:

$$\begin{equation} \begin{split} &\left\|\sum _{i_1+i_2+i_3\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3+1}\eta \right\|_{L^1_{\underline{u}}L^2(S)} \\ &\qquad \leq C\epsilon \left(\sum _{i_1\leq 3}\sum _{i_2\leq 4}||\nabla ^{i_1}\psi ||^{i_2}_{L^{{\infty }}_{\underline{u}}L^2(S)}\right)+C\epsilon ^{\frac{1}{2}}f(u)^{-1}||f(u)\nabla ^4\eta ||_{L^2_{\underline{u}}L^2(S)}\\ &\qquad \leq C\epsilon (1+\Delta _1)^4+C\epsilon ^{\frac{1}{2}}f(u)^{-1}\tilde{\mathcal{O}}_{4,2}. \end{split} {\tag{44}} \end{equation}$$

We now move to control the terms that are nonlinear in the Ricci coefficients. First, we estimate the terms without $\psi _H$ or $\psi _{\underline{H}}$:

$$\begin{equation} \begin{split} &\left\|\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi \right\|_{L^1_{\underline{u}}L^2(S)} \\ &\qquad \leq C\epsilon \left(\sum _{i_1\leq 1,i_2\leq 4}||\nabla ^{i_1}\psi ||_{L^\infty _{\underline{u}} L^\infty (S)}^{i_2}\right)\left(\sum _{i_3\leq 3}||\nabla ^{i_3}\psi ||_{ L^\infty _{\underline{u}} L^2(S)}\right)\\ &\qquad \leq C\epsilon (1+\Delta _1)^5. \end{split} {\tag{45}} \end{equation}$$

We then control the term with both $\psi _H$ and $\psi _{\underline{H}}$:

$$\begin{equation} \begin{split} &\left\|\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}\right\|_{L^1_{\underline{u}}L^2(S)} \\ &\ \leq C\left(\sum _{i_1\leq 1,i_2\leq 3}\!||\nabla ^{i_1}\psi ||_{L^\infty _{\underline{u}}L^\infty (S)}^{i_2}\!\right)\!\!\left(\sum _{i_3\leq 1}\!||\nabla ^{i_3}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^\infty (S)}\!\right)\!\!\left(\sum _{i_4\leq 3}\!||\nabla ^{i_4}\psi _{H}||_{L^1_{\underline{u}}L^2(S)}\!\right)\\ &\quad +C\left(\!\sum _{i_1\leq 1,i_2\leq 3}\!||\nabla ^{i_1}\psi ||_{L^\infty _{\underline{u}}L^\infty (S)}^{i_2}\!\!\right)\!\!\left(\sum _{i_3\leq 3}\!||\nabla ^{i_3}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2(S)}\!\!\right)\!\!\left(\sum _{i_4\leq 1}\!||\nabla ^{i_4}\psi _{H}||_{L^1_{\underline{u}}L^\infty (S)}\!\!\right) \\ &\quad +C||\nabla ^2\psi ||_{L^\infty _{\underline{u}}L^2(S)}||\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^\infty (S)}||\psi _{H}||_{L^1_{\underline{u}}L^\infty (S)} \\ &\ \leq C\left(\sum _{i_1\leq 1,i_2\leq 3}||\nabla ^{i_1}\psi ||_{L^\infty _{\underline{u}}L^\infty (S)}^{i_2}\right)\left(\sum _{i_3\leq 1}||\nabla ^{i_3}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^\infty (S)}\right)\\ &\quad \quad \times \left(\sum _{i_4\leq 3}||f(\underline{u})\nabla ^{i_4}\psi _{H}||_{L^2_{\underline{u}}L^2(S)}\right)||f(\underline{u})^{-1}||_{L^2_{\underline{u}}} \\ &\quad +C\left(\sum _{i_1\leq 1,i_2\leq 3}||\nabla ^{i_1}\psi ||_{L^\infty _{\underline{u}}L^\infty (S)}^{i_2}\right)\left(\sum _{i_3\leq 3}||\nabla ^{i_3}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2(S)}\right)\\ &\quad \quad \times \left(\sum _{i_4\leq 1}||f(\underline{u})\nabla ^{i_4}\psi _{H}||_{L^2_{\underline{u}}L^\infty (S)}\right)||f(\underline{u})^{-1}||_{L^2_{\underline{u}}} \\ &\quad +C||\nabla ^2\psi ||_{L^\infty _{\underline{u}}L^2(S)}||\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^\infty (S)}||f(\underline{u})\psi _{H}||_{L^2_{\underline{u}}L^\infty (S)}||f(\underline{u})^{-1}||_{L^2_{\underline{u}}} \\ &\ \leq C\epsilon (1+\Delta _1)^3\left(\sum _{i_1\leq 3}||f(\underline{u})\nabla ^{i_1}\psi _H||_{L^2_{\underline{u}}L^2(S)}\right)\left(\sum _{i_2\leq 3}||\nabla ^{i_2}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2(S)}\right)\\ &\ \leq C\epsilon \Delta _1(1+\Delta _1)^3\left(\sum _{i\leq 3}||\nabla ^i\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2(S)}\right). \end{split} {\tag{46}} \end{equation}$$

Therefore, by the bounds 43, 44, 45, and 46, we have that for every fixed $u$,

$$\begin{equation*} \begin{split} &\sum _{i\leq 3}||\nabla ^i\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^{{2}}(S)}\\ &\qquad \leq {C\left(\sum _{i\leq 3}||\nabla ^i\psi _{\underline{H}}||_{L^2(S_{u,0})}\right)}+C\epsilon ^{\frac{1}{2}}f(u)^{-1}(\mathcal{R}+\tilde{\mathcal{O}}_{4,2})+C\epsilon (1+\Delta _1)^5\\ &\qquad \quad +C\epsilon \Delta _1(1+\Delta _1)^3\left(\sum _{i\leq 3}||\nabla ^i\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2(S)}\right). \end{split} \end{equation*}$$

We now multiply this inequality by $f(u)$ and take the $L^2$ norm in $u$ to get

$$\begin{equation*} \begin{split} &\sum _{i\leq 3}||f(u)\nabla ^i\psi _{\underline{H}}||_{L^2_{u}L^\infty _{\underline{u}}L^2(S)} \\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )+C\epsilon ^{\frac{1}{2}}||f(u)(f(u)^{-1})||_{L^2_u}(\mathcal{R}+\tilde{\mathcal{O}}_{4,2})+C\epsilon (1+\Delta _1)^5\\ &\qquad \quad +C\epsilon \Delta _1(1+\Delta _1)^3\left(\sum _{i\leq 3}||f(u)\nabla ^i\psi _{\underline{H}}||_{L^2_uL^\infty _{\underline{u}}L^2(S)}\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} ), \end{split} \end{equation*}$$

for $\epsilon$ sufficiently small.

■

Using instead the equation for $\nabla _3\psi _H$, we obtain the following estimates in a completely analogous manner:

Proposition 14.

Assume

$$\mathcal{R} <\infty ,\qquad \tilde{\mathcal{O}}_{4,2}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\Delta _1, \mathcal{R},\tilde{\mathcal{O}}_{4,2})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 3}\mathcal{O}_{i,2}[\psi _{H}]\leq C(\mathcal{O}_{\mathrm{ini}} ) .$$

In particular this estimate is independent of $\Delta _1$.

By the Sobolev embedding theorems given by Proposition 7, we have thus closed our bootstrap assumption (A1) after choosing $\Delta _1$ to be sufficiently large depending on the initial data norm $\mathcal{O}_{\mathrm{ini}}$. We have therefore proved the desired estimates for the Ricci coefficients and their first three angular covariant derivatives. We summarize this in the following proposition.

Proposition 15.

Assume

$$\mathcal{R} <\infty ,\qquad \tilde{\mathcal{O}}_{4,2}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R},\tilde{\mathcal{O}}_{4,2})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 3}\mathcal{O}_{i,2}[\psi ,\psi _{\underline{H}},\psi _H]\leq C(\mathcal{O}_{\mathrm{ini}} ).$$

7. Elliptic estimates for fourth derivatives of the Ricci coefficients

We now estimate the fourth derivative of the Ricci coefficients. We introduce the following bootstrap assumption:

$$\begin{equation} \tilde{\mathcal{O}}_{4,2}\leq \Delta _2, {\tag{A2}} \end{equation}$$

where $\Delta _2$ is a constant to be chosen later.

The estimates for the fourth derivative of the Ricci coefficients cannot be achieved only by the transport equations since there would be a loss in derivatives. We can however use the transport equation—the Hodge system type estimates as in 51516. We will first derive estimates for some chosen combination of $\nabla ^4(\psi ,\psi _H,\psi _{\underline{H}})+{\nabla ^3}(\beta ,K,\check{\sigma },\underline{\beta })$ by using transport equations. We will then show that the estimates for all the fourth derivatives of the Ricci coefficients can be proved via elliptic estimates.

In order to apply the elliptic estimates in section 5.5, we need to first control the Gauss curvature and its first and second derivatives in $L^2(S)$.

Proposition 16.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 2}||\nabla ^i K||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} {)}.$$

Proof.

$K$ obeys the following Bianchi equation:

$$\nabla _4 K=\nabla \beta +\psi _{H}K+\sum _{i_1+i_2+i_3\leq 1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _H.$$

Commuting with angular derivatives, we have, for $i\leq 2$,

$$\begin{equation*} \begin{split} &\nabla _4\nabla ^i K\\ &\qquad =\sum _{i_1+i_2+i_3\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3+1}\beta +\sum _{i_1+i_2+i_3+i_4\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}K\\ &\qquad \quad +\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _H. \end{split} \end{equation*}$$

By Proposition 5, in order to control $\nabla ^iK$ in $L^\infty _uL^\infty _{\underline{u}}L^2(S)$, we need to bound the right hand side in $L^\infty _uL^1_{\underline{u}}L^2(S)$. We first control the term containing $\beta$:

$$\begin{equation*} \begin{split} &\left\|\sum _{i_1+i_2+i_3\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3+1}\beta \right\|_{L^\infty _uL^1_{\underline{u}}L^2(S)}\\ &\quad \leq C\!\left(\sum _{i_1\leq 1,i_2\leq 2}\!||\nabla ^{i_1}\psi ||_{L^\infty _uL^\infty _{\underline{u}}L^\infty (S)}^{i_2}\!\right)||f(\underline{u})^{-1}||_{L^2_{\underline{u}}}\left(\sum _{i_3\leq 2}||f(\underline{u})\nabla ^{i_2+1}\beta ||_{L^\infty _u L^2_{\underline{u}} L^2(S)}\!\!\right)\\ &\quad \leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon \mathcal{R}, \end{split} \end{equation*}$$

where we have used the estimates for $\psi$ given by Proposition 15. The term containing $K$ can be controlled by

$$\begin{equation*} \begin{split} &\left\|\sum _{i_1+i_2+i_3+i_4\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}K\right\|_{L^\infty _uL^1_{\underline{u}}L^2(S)}\\ &\ \leq C\left(\sum _{i_1\leq 1,i_2\leq 2}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _uL^\infty _{\underline{u}}L^\infty (S)}\right)\int _0^{\underline{u}}\left(\sum _{i_3+i_4\leq 2}||\nabla ^{i_3}\psi _H\nabla ^{i_4}K||_{L^\infty _u L^2(S_{u,\underline{u}'})}\right)d\underline{u}'\\ &\ \leq C(\mathcal{O}_{\mathrm{ini}} )\int _0^{\underline{u}}\left(\sum _{i_1+i_2\leq 2}||\nabla ^{i_1}\psi _H\nabla ^{i_2}K||_{L^\infty _u L^2(S_{u,\underline{u}'})}\right)d\underline{u}'\\ &\ \leq C(\mathcal{O}_{\mathrm{ini}} )\int _0^{\underline{u}}\left(\sum _{i_1\leq 2}||\nabla ^{i_1}\psi _H||_{L^\infty _u L^2(S_{u,\underline{u}'})}\right)\left(\sum _{i_2\leq 2}||\nabla ^{i_2}K||_{L^\infty _u L^2(S_{u,\underline{u}'})}\right)d\underline{u}'. \end{split} \end{equation*}$$

The remaining term has been bounded in the previous section. By 42 and Proposition 15,

$$\begin{equation*} \left\|\sum _{i_1+i_2+i_3\leq 2}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _H\right\|_{L^\infty _uL^1_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon . \end{equation*}$$

Therefore, by Proposition 5,

$$\begin{equation*} \begin{split} \sum _{i\leq 2}||\nabla ^i K||_{L^\infty _u L^2(S_{u,\underline{u}})} &\leq C(\mathcal{O}_{\mathrm{ini}} )\left(1\!+\!\epsilon \!+\!\epsilon \mathcal{R}\!+\!\!\int _0^{\underline{u}}\!\left(\sum _{i_1\leq 2}\!||\nabla ^{i_1}\psi _H||_{L^\infty _u L^2(S_{u,\underline{u}'})}\right)\right.\\ &\qquad \qquad \qquad \times \left.\left(\sum _{i_2\leq 2}\!||\nabla ^{i_2}K||_{L^\infty _u L^2(S_{u,\underline{u}'})}\right)d\underline{u}'\right). \end{split} \end{equation*}$$

Gronwall’s inequality implies

$$\begin{equation*} \begin{split} \sum _{i\leq 2}||\nabla ^i K||_{L^\infty _u L^2(S_{u,\underline{u}})} \leq C(\mathcal{O}_{\mathrm{ini}} )\exp \left(\sum _{i\leq 2}||\nabla ^i\psi _H||_{L^1_{\underline{u}} L^\infty _u L^2(S)}\right) \leq C(\mathcal{O}_{\mathrm{ini}} ) \end{split} \end{equation*}$$

since by Proposition 15, $\sum _{i\leq 1}||\nabla ^i\psi _H||_{L^1_{\underline{u}} L^\infty _u L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )$ for $\epsilon$ sufficiently small.

■

It is easy to see that since $\check{\sigma }$ satisfies a similar schematic Bianchi equation as $K$, we also have the following estimates for $\check{\sigma }$ and its derivative.

Proposition 17.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 2}||\nabla ^i\check{\sigma }||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} ).$$

Using Proposition 16, we now control the fourth derivatives of the Ricci coefficients. We first bound $\nabla ^4\mathrm{tr\,}\chi$ using the transport equation.

Proposition 18.

There exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2)$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(\underline{u})\nabla ^4\mathrm{tr\,}\chi ||_{L^2_{\underline{u}}L^\infty _uL^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} ).$$

Proof.

Consider the following equation:

$$\nabla _4 \mathrm{tr\,}\chi =\psi _H\psi _H,$$

After commuting with angular derivatives, we have

$$\nabla _{4}\nabla ^{4}\mathrm{tr\,}\chi =\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _H.$$

By Proposition 5, in order to control $\nabla ^4\mathrm{tr\,}\chi$ in $L^2(S_{u,\underline{u}})$, we need to bound the right-hand side in $L^1_{\underline{u}} L^2(S)$. Using the fact that $f(\underline{u})$ is decreasing, this can be achieved using Sobolev embedding (Proposition 7) by

$$\begin{equation*} \begin{split} &{\sum _{i_1+i_2+i_3+i_4\leq 4}\int _0^{\underline{u}}} ||\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _H||_{L^2(S_{u,\underline{u}'})} {d\underline{u}'}\\ &\qquad \leq Cf(\underline{u})^{-2}\left(\sum _{i_1\leq 3}\sum _{i_2\leq 4}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{\underline{u}}L^2(S)}\right)\left(\sum _{i_3\leq 2}||f(\underline{u})\nabla ^{i_3}\psi _H||_{L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \quad \times \left(\sum _{i_4\leq 4}||f(\underline{u})\nabla ^{i_4}\psi _H||_{L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \leq Cf(\underline{u})^{-2}\Delta _2. \end{split} \end{equation*}$$

By Proposition 5, we have

$$\begin{equation} ||\nabla ^4\mathrm{tr\,}\chi ||_{L^2(S_{u,\underline{u}})}\leq C(\mathcal{O}_{\mathrm{ini}} )+C(\mathcal{O}_{\mathrm{ini}} ) f(\underline{u})^{-2}\Delta _2. {\tag{47}} \end{equation}$$

Multiplying 47 by $f(\underline{u})$ and taking first the $L^\infty$ norm in $u$ and then the $L^2$ norm in $\underline{u}$, we have

$$\begin{equation*} \begin{split} ||f(\underline{u})\nabla ^{4}\mathrm{tr\,}\chi ||_{L^2_{\underline{u}}L^\infty _u L^2(S)}\leq &C(\mathcal{O}_{\mathrm{ini}} )+C(\mathcal{O}_{\mathrm{ini}} )||f(\underline{u})^{-1}||_{L^2_{\underline{u}}}\Delta _2\leq C(\mathcal{O}_{\mathrm{ini}} )+C\epsilon \Delta _2, \end{split} \end{equation*}$$

where we have used

$$||f(\underline{u})^{-1}||_{L^2_{\underline{u}}}\leq C\epsilon .$$

Thus, the conclusion follows by choosing $\epsilon$ to be sufficiently small depending on $\Delta _2$.

■

Once we have the estimates for $\nabla ^4\mathrm{tr\,}\chi$, we can control $\nabla ^4\hat{\chi }$ using elliptic estimates:

Proposition 19.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(\underline{u})\nabla ^4\hat{\chi }||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )+C\mathcal{R}.$$

Proof.

We now use the Codazzi equation

$$\mathrm{div\,}\hat{\chi }=\frac{1}{2}\nabla \mathrm{tr\,}\chi -\beta +\psi \psi _H$$

and apply elliptic estimates from Proposition 11 to get

$$\begin{equation} \begin{split} ||\nabla ^4\hat{\chi }||_{L^2(S)} \leq &C\left(\sum _{i\leq 4}||\nabla ^i\mathrm{tr\,}\chi ||_{L^2(S)}+\sum _{i\leq 3}||\nabla ^i\beta ||_{L^2(S)}\right.\\ &\qquad \left.+\sum _{i_1+i_2\leq 3}||\nabla ^{i_1}\psi \nabla ^{i_2}\psi _H||_{L^2(S)} +\sum _{i\leq 3}||\nabla ^i\hat{\chi }||_{L^2(S)}\right). \end{split} {\tag{48}} \end{equation}$$

Notice that we can apply elliptic estimates using Proposition 11, since we have the estimates for the Gauss curvature from Proposition 16. Multiply (48) by $f(\underline{u})$ and take the $L^\infty _uL^2_{\underline{u}}$ norm to get

$$\begin{equation*} \begin{split} &||f(\underline{u})\nabla ^4\hat{\chi }||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\\ &\qquad \leq C\left(\sum _{i\leq 4}||f(\underline{u})\nabla ^i\mathrm{tr\,}\chi ||_{L^\infty _uL^2_{\underline{u}}L^2({S})}+\sum _{i\leq 3}||f(\underline{u})\nabla ^i\beta ||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\right.\\ &\qquad \quad \left.+\sum _{i_1+i_2\leq 3}||f(\underline{u})\nabla ^{i_1}\psi \nabla ^{i_2}\psi _H||_{L^\infty _uL^2_{\underline{u}}L^2(S)} +\sum _{i\leq 3}||f(\underline{u})\nabla ^i\hat{\chi }||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )+C\mathcal{R}+C\sum _{i_1+i_2\leq 3}||f(\underline{u})\nabla ^{i_1}\psi \nabla ^{i_2}\psi _H||_{L^\infty _uL^2_{\underline{u}}L^2(S)}. \end{split} \end{equation*}$$

By Proposition 15 and Sobolev embedding theorem in Proposition 7, we have

$$\begin{equation*} \begin{split} &\sum _{i_1+i_2\leq 3}||f(\underline{u})\nabla ^{i_1}\psi \nabla ^{i_2}\psi _H||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\\ &\qquad \leq C\left(\sum _{i_1\leq 3}||\nabla ^{i_1}\psi ||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}\right)\left(\sum _{i_2\leq 3}||f(\underline{u})\nabla ^{i_2}\psi _H||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\right)\leq C(\mathcal{O}_{\mathrm{ini}} ). \end{split} \end{equation*}$$

Therefore,

$$\begin{equation*} ||f(\underline{u})\nabla ^4\hat{\chi }||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )+C\mathcal{R}. \end{equation*}$$■

The $\tilde{\mathcal{O}}_{4,2}$ estimates for $\nabla ^4\mathrm{tr\,}\underline{\chi }$ and $\nabla ^4\hat{\underline{\chi }}$ follow identically as that for $\nabla ^4\mathrm{tr\,}\chi$ and $\nabla ^4\hat{\chi }$:

Proposition 20.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(u)\nabla ^4\mathrm{tr\,}\underline{\chi }||_{L^\infty _{\underline{u}}L^2_{u}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )$$

and

$$||f(u)\nabla ^4\hat{\underline{\chi }}||_{L^\infty _{\underline{u}}L^2_{u}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )+C\mathcal{R}.$$

We then prove estimates for $\nabla ^4\eta$. To do so, we first prove estimates for third derivatives of $\mu =-\mathrm{div\,}\eta +K$ and recover the control for $\nabla ^4\eta$ via elliptic estimates.

Proposition 21.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(u)\nabla ^4\eta ||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\mathcal{R})$$

and

$$||f(\underline{u})\nabla ^4\eta ||_{L^\infty _{\underline{u}}L^2_{u}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\mathcal{R}).$$

Proof.

Recall that

$$\mu =-\mathrm{div\,}\eta +K.$$

Then $\mu$ satisfies the following equation:⁠²⁶

²⁶

It is important to note that the potentially harmful term $\psi _{\underline{H}}\psi _H\psi _H$ is absent in this equation. This required structure is the reason that we perform this renormalization instead of using $\mu =-\mathrm{div\,}\eta -\rho +\frac{1}{2}\hat{\chi }\cdot \hat{\underline{\chi }}$ as in 1819.

✖

$$\nabla _4 \mu =\psi _H(K,\check{\sigma })+\sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _H.$$

After commuting with angular derivatives, we get

$$\begin{equation*} \begin{split} \nabla _{4}\nabla ^3\mu =&\sum _{i_1+i_2+i_3+i_4=3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}(K,\check{\sigma })+\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _H. \end{split} \end{equation*}$$

We now control each of the terms on the right-hand side in $L^1_{\underline{u}}L^2(S)$. The first term, which contains curvature components, can be estimated by

$$\begin{equation*} \begin{split} &{\sum _{i_1+i_2+i_3+i_4=3}\int _0^{\underline{u}}}||\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}(K,\check{\sigma })||_{L^2(S_{u,\underline{u}'})}{d\underline{u}'}\\ &\qquad \leq Cf(u)^{-1}f(\underline{u})^{-1}\left(\sum _{i_1\leq 3}\sum _{i_2\leq 3}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{\underline{u}}L^2(S)}\right)\left(\sum _{i_3\leq 3}||f(\underline{u})\nabla ^{i_3}\psi _H||_{L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \qquad \times \left(\sum _{i_4\leq 3}||f(u)\nabla ^{i_4}(K,\check{\sigma })||_{L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )f(u)^{-1}f(\underline{u})^{-1}\mathcal{R}, \end{split} \end{equation*}$$

using the bounds obtained in Proposition 15. The second term can be controlled using Sobolev embedding in Proposition 7 by

$$\begin{equation*} \begin{split} &{\sum _{i_1+i_2+i_3+i_4=4}\int _0^{\underline{u}}}||\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _H||_{L^2(S_{u,\underline{u}'})}{d\underline{u}'}\\ &\qquad \leq Cf(u)^{-1}f(\underline{u})^{-1}\left(\sum _{i_1\leq 4}\sum _{i_2\leq 5}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{\underline{u}}L^2(S)}\right)\left(\sum _{i_3\leq 4}||f(\underline{u})\nabla ^{i_3}\psi _H||_{L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \qquad \times \left(\sum _{i_4\leq 4}||f(u)\nabla ^{i_4}\psi ||_{L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )f(u)^{-1}f(\underline{u})^{-1}(1+\Delta _2)^2 \end{split} \end{equation*}$$

using the estimates in Proposition 15. Therefore, by Proposition 5, we have

$$\begin{equation} \begin{split} ||\nabla ^3\mu ||_{L^2(S_{u,\underline{u}})} \leq &C(\mathcal{O}_{\mathrm{ini}} )(1+ f(u)^{-1}f(\underline{u})^{-1}(\mathcal{R}+(1+\Delta _2)^2)). \end{split} {\tag{49}} \end{equation}$$

Recall that the $L^2_{\underline{u}}$ norm of $f(\underline{u})^{-1}$ is bounded by $\epsilon$. Thus, multiplying 49 by $f(u)$ and taking the $L^2$ norm in $\underline{u}$, we get

$$\begin{equation*} \begin{split} ||f(u)\nabla ^3\mu ||_{L^2_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\epsilon (\mathcal{R}+(1+\Delta _2)^2))\leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon ^{\frac{1}{2}} \end{split} \end{equation*}$$

for $\epsilon$ sufficiently small. Similarly, multiplying 49 by $f(\underline{u})$ and taking the $L^2$ norm in $u$, we get

$$||f(\underline{u})\nabla ^3\mu ||_{L^2_{u}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon ^{\frac{1}{2}}.$$

We can obtain bounds for $\nabla ^4\eta$ from the control of $\nabla ^3\mu$ using elliptic estimates as follows. By the div-curl systems

$$\mathrm{div\,}\eta =-\mu +K,\qquad \mathrm{curl\,}\eta =\check{\sigma },$$

and the elliptic estimates given by Propositions 10 and 16, we have

$$\begin{equation*} ||\nabla ^4\eta ||_{L^2(S)}\leq C\left(\sum _{i\leq 3}||\nabla ^i\mu ||_{L^2(S)} +\sum _{i\leq 3}||\nabla ^i(K,\check{\sigma })||_{L^2(S)}+\sum _{i\leq 3}||\nabla ^i\eta ||_{L^2(S)}\right). \end{equation*}$$

Therefore,

$$\begin{equation*} \begin{split} &||f(u)\nabla ^4\eta ||_{L^2_{\underline{u}}L^2(S)}\\ &\ \leq C\left(\sum _{i\leq 3}\!||f(u)\nabla ^i\mu ||_{L^2_{\underline{u}}L^2(S)}\!+\!\sum _{i\leq 3}\!||f(u)\nabla ^i(K,\check{\sigma })||_{L^2_{\underline{u}}L^2(S)}\!+\!\sum _{i\leq 3}\!||f(u)\nabla ^i\eta ||_{L^2_{\underline{u}}L^2(S)}\!\right)\\ &\ \leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\mathcal{R}). \end{split} \end{equation*}$$

Similarly,

$$\begin{equation*} ||f(\underline{u})\nabla ^4\eta ||_{L^2_{u}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\mathcal{R}). \end{equation*}$$■

A similar proof shows that the conclusion of Proposition 21 holds also for $\nabla ^3\underline{\eta }$:

Proposition 22.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(u)\nabla ^4\underline{\eta }||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\mathcal{R})$$

and

$$||f(\underline{u})\nabla ^4\underline{\eta }||_{L^\infty _{\underline{u}}L^2_{u}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(\epsilon ^{\frac{1}{2}}+\mathcal{R}).$$

We now move to the estimates for $\nabla ^4\underline{\omega }$:

Proposition 23.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(u)\nabla ^4\underline{\omega }||_{L^\infty _{\underline{u}}L^2_uL^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(1+\mathcal{R}).$$

Proof.

Let $\underline{\omega }^{\dagger }$ be defined as the solution to

$$\nabla _4\underline{\omega }^{\dagger }=\frac{1}{2} \check{\sigma }$$

with zero data on $\underline{H}_0$ and

$$\underline{\kappa }:=-\nabla \underline{\omega }+^*\nabla \underline{\omega }^{\dagger }-\frac{1}{2}\underline{\beta }.$$

By the definition of $\underline{\omega }^{\dagger }$, it is easy to see that using Proposition 5,

$$\sum _{i\leq 3}||\nabla ^i\underline{\omega }^{\dagger }||_{L^2_uL^\infty _{\underline{u}}L^2(S)}\leq C\epsilon \mathcal{R}\leq C(\mathcal{O}_{\mathrm{ini}} ).$$

In other words, ${\nabla ^i}\underline{\omega }^{\dagger }$ satisfies much better estimates⁠²⁷ than ${\nabla ^i}\psi _{\underline{H}}$ for $i\leq 3$. With this in mind, in the proof of this proposition, we will also use $\psi _{\underline{H}}$ to denote $\underline{\omega }^{\dagger }$ (in addition to $\mathrm{tr\,}\underline{\chi }$, $\hat{\underline{\chi }},$ and $\underline{\omega }$).

²⁷

We recall that for $\psi _{\underline{H}}$ we only have the degenerate estimate

$$\sum _{i\leq 3}||f(u)\nabla ^i\psi _{\underline{H}}||_{L^2_uL^\infty _{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} ).$$

✖

With this convention, $\underline{\kappa }$ then obeys the schematic equation

$$\nabla _4 \underline{\kappa }=\psi (K,\check{\sigma })+\sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi +\sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi _H\nabla ^{i_3}\psi _{\underline{H}}.$$

After commuting with angular derivatives, we get

$$\begin{equation*} \begin{split} \nabla _{4}\nabla ^3\underline{\kappa }&=\sum _{i_1+i_2+i_3+i_4=3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma }) +\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi \\ &\quad +\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}.\\ \end{split} \end{equation*}$$

Therefore,

$$\begin{equation*} \begin{split} ||\nabla ^3\underline{\kappa }||_{L^2(S_{u,\underline{u}})} &\leq C||\nabla ^3\underline{\kappa }||_{L^2(S_{u,0})}\!+\!C||\!\sum _{i_1+i_2+i_3+i_4=3}\!\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })||_{L^1_{\underline{u}}L^2(S)}\\ &\quad +C||\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi ||_{L^1_{\underline{u}}L^2(S)}\\ &\quad +C||\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}||_{L^1_{\underline{u}}L^2(S)}. \end{split} \end{equation*}$$

Multiplying by $f(u)$ and taking the $L^2$ norm in $u$, we get

$$\begin{equation*} \begin{split} &||f(u)\nabla ^3\underline{\kappa }||_{L^2_uL^2(S)}\\ &\ \leq C||f(u)\nabla ^3\underline{\kappa }||_{L^2_uL^2(S_{u,0})}+C||f(u)\!\sum _{i_1+i_2+i_3+i_4=3}\!\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })||_{{L^2_u}L^1_{\underline{u}}L^2(S)}\\ &\quad +C||f(u)\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi ||_{L^2_uL^1_{\underline{u}}L^2(S)}\\ &\quad +C||f(u)\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}||_{L^2_uL^1_{\underline{u}}L^2(S)}. \end{split} \end{equation*}$$

The first term is an initial data term and it is bounded by a constant depending only on $\mathcal{O}_{\mathrm{ini}}$. We estimate each of the nonlinear terms. The second term can be controlled by

$$\begin{equation*} \begin{split} &||f(u)\sum _{i_1+i_2+i_3+i_4=3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })||_{L^2_uL^1_{\underline{u}}L^2(S)}\\ &\qquad \leq C\epsilon \left(\sum _{i_1\leq 3}\sum _{i_2\leq 4}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{\underline{u}}L^\infty _{\underline{u}}L^2(S)}\right)\left(\sum _{i_3\leq 3}||f(u)\nabla ^{i_3}(K,\check{\sigma })||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon \mathcal{R}. \end{split} \end{equation*}$$

The third term can be bounded by

$$\begin{equation*} \begin{split} &||f(u)\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi ||_{L^2_uL^1_{\underline{u}}L^2(S)}\\ &\qquad \leq C\epsilon \left(\sum _{i_1\leq 3}\sum _{i_2\leq 4}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{\underline{u}}L^\infty _uL^2(S)}\right)\left(\sum _{i_3\leq 3}||\nabla ^{i_3}\psi ||_{L^\infty _{\underline{u}}L^\infty _uL^2(S)}\right)\\ &\qquad \quad \times \left(\sum _{i_4\leq 4}||f(u)\nabla ^{i_4}\psi ||_{L^\infty _{u}L^2_{\underline{u}}L^2(S)}\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon \Delta _2. \end{split} \end{equation*}$$

The fourth term can be estimated by

$$\begin{equation*} \begin{split} &||f(u)\sum _{i_1+i_2+i_3+i_4=4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}||_{L^2_uL^1_{\underline{u}}L^2(S)}\\ &\qquad \leq C\epsilon ^{\frac{1}{2}}\left(\sum _{i_1\leq 3}\sum _{i_2\leq 4}||\nabla ^{i_1}\psi ||^{i_2}_{L^\infty _{\underline{u}}L^\infty _uL^2(S)}\right)\left(\sum _{i_3\leq 3}||\nabla ^{i_3}\psi _H||_{L^1_{\underline{u}}L^\infty _uL^2(S)}\right)\\ &\qquad \quad \times \left(\sum _{i_4\leq 4}||f(u)\nabla ^{i_4}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2_uL^2(S)}\right)\\ &\qquad \quad +C||f(\underline{u})^{-1}||_{L^2_{\underline{u}}}||f(\underline{u})\nabla ^4\psi _H||_{L^\infty _uL^2_{\underline{u}}L^2(S)}||f(u)\psi _{\underline{H}}||_{L^2_uL^\infty _{\underline{u}} L^\infty (S)}\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )\epsilon (1+\Delta _2). \end{split} \end{equation*}$$

Therefore,

$$\begin{equation} ||{f(u)}\nabla ^3\underline{\kappa }||_{L^2_uL^\infty _{\underline{u}} L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(1+\epsilon (1+\Delta _2+\mathcal{R}))\leq C(\mathcal{O}_{\mathrm{ini}} ),{\tag{50}} \end{equation}$$

after choosing $\epsilon$ to be sufficiently small. Finally, we retrieve the estimates for $\nabla ^4\underline{\omega }$ and $\nabla ^4\underline{\omega }^{\dagger }$ from the bounds for $\nabla ^3\underline{\kappa }$. To this end, consider the div-curl system

$$\mathrm{div\,}\nabla \underline{\omega }=-\mathrm{div\,}\underline{\kappa }-\frac{1}{2}\mathrm{div\,}\underline{\beta },$$

$$\mathrm{curl\,}\nabla \underline{\omega }=0,$$

$$\mathrm{div\,}\nabla \underline{\omega }^{\dagger }=-\mathrm{curl\,}\underline{\kappa }-\frac{1}{2}\mathrm{curl\,}\underline{\beta },$$

$$\mathrm{curl\,}\nabla \underline{\omega }^{\dagger }=0.$$ By elliptic estimates given by Propositions 10 and 16, we have

$$\begin{equation*} \begin{split} &||\nabla ^4(\underline{\omega },\underline{\omega }^{\dagger })||_{L^2(S_{u,\underline{u}})}\\ &\qquad \leq C\left(\sum _{i\leq 3}||\nabla ^i\underline{\kappa }||_{L^2(S_{u,\underline{u}})}+\sum _{i\leq 3}||\nabla ^i\underline{\beta }||_{L^2(S_{u,\underline{u}})}+\sum _{i\leq 3}||\nabla ^i(\underline{\omega },\underline{\omega }^{\dagger })||_{L^2(S_{u,\underline{u}})}\right). \end{split} \end{equation*}$$

Therefore, using Proposition 12, 50, and the curvature norm,

$$\begin{equation*} ||{f(u)}\nabla ^4(\underline{\omega },\underline{\omega }^{\dagger })||_{L^\infty _{\underline{u}}L^2_uL^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(1+\mathcal{R}).{} \end{equation*}$$■

By switching $\underline{\omega }$ and $\omega$ as well as $e_3$ and $e_4$, we also have the following estimates for $\nabla ^4\omega$:

Proposition 24.

Assume

$$\mathcal{R}<\infty .$$

Then there exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\Delta _2,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$||f(\underline{u})\nabla ^4\omega ||_{L^\infty _{u}L^2_{\underline{u}}L^2(S)}\leq C(\mathcal{O}_{\mathrm{ini}} )(1+\mathcal{R}).$$

We have thus controlled the fourth angular derivatives of all Ricci coefficients and have closed the bootstrap assumption (A2) after choosing $\Delta _2$ to be sufficiently large depending on $\mathcal{O}_{\mathrm{ini}}$ and $\mathcal{R}$. We summarize this in the following proposition:

Proposition 25.

Assume

$$\mathcal{R}<\infty .$$

There exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R})$ such that whenever $\epsilon \leq \epsilon _0$,

$$\tilde{\mathcal{O}}_{4,2}\leq C(\mathcal{O}_{\mathrm{ini}} )(1+\mathcal{R}).$$

8. Estimates for curvature

In this section, we derive and prove the energy estimates. To this end, we introduce the following bootstrap assumptions:

$$\begin{equation} \mathcal{R}\leq \Delta _3, {\tag{A3}} \end{equation}$$

where $\Delta _3$ is a constant to be chosen later.

In order to derive the energy estimates, we need the following integration by parts formula, which can be proved by direct computation:

Proposition 26.

Let $D_{u,\underline{u}}$ be defined as the spacetime region whose coordinates $(u',\underline{u}')$ satisfy $0\leq u'\leq u$ and $0\leq \underline{u}'\leq \underline{u}$. Suppose $\phi _1$ and $\phi _2$ are tensors of rank $r$, then

$$\int _{D_{u,\underline{u}}} \phi _1 \nabla _4\phi _2+\int _{D_{u,\underline{u}}}\phi _2\nabla _4\phi _1= \int _{\underline{H}_{\underline{u}}(0,u)} \phi _1\phi _2-\int _{\underline{H}_0(0,u)} \phi _1\phi _2+\int _{D_{u,\underline{u}}}(2\omega -\mathrm{tr\,}\chi )\phi _1\phi _2,$$

$$\int _{D_{u,\underline{u}}} \phi _1 \nabla _3\phi _2+\int _{D_{u,\underline{u}}}\phi _2\nabla _3\phi _1= \int _{H_{u}(0,\underline{u})} \phi _1\phi _2-\int _{H_0(0,\underline{u})} \phi _1\phi _2+\int _{D_{u,\underline{u}}}(2\underline{\omega }-\mathrm{tr\,}\underline{\chi })\phi _1\phi _2.$$

Proposition 27.

Suppose we have a tensor $^{(1)}\phi$ of rank $r$ and a tensor $^{(2)}\phi$ of rank $r-1$. Then

$$\begin{equation*} \begin{split} &\int _{D_{u,\underline{u}}}{ }^{(1)}\phi ^{A_1A_2\cdots A_r}\nabla _{A_r}{ }^{(2)}\phi _{A_1\cdots A_{r-1}}+\int _{D_{u,\underline{u}}}\nabla ^{A_r}{ }^{(1)}\phi _{A_1A_2\cdots A_r}{ }^{(2)}\phi ^{A_1\cdots A_{r-1}}\\ &\qquad = -\int _{D_{u,\underline{u}}}(\eta +\underline{\eta }){ }^{(1)}\phi { }^{(2)}\phi . \end{split} \end{equation*}$$

With these we are now ready to derive energy estimates for $\nabla ^i(K,\check{\sigma })$ in $L^2(H_u)$ and for $\nabla ^i\underline{\beta }$ in $L^2(\underline{H}_{\underline{u}})$. The most important observation is that the two uncontrollable terms have favorable signs. This in turn is due to the choice of $f(u)$ which is decreasing toward the future.

Proposition 28.

The following $L^2$ estimates for the curvature hold:

$$\begin{equation*} \begin{split} &\sum _{i\leq 3}(||f(u)\nabla ^i(K,\check{\sigma })||^2_{L^\infty _u L^2_{\underline{u}}L^2(S)}+||f(u)\nabla ^i\underline{\beta }||^2_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}) \\ &\qquad \leq \sum _{i\leq 3}(||f(u)\nabla ^i(K,\check{\sigma })||^2_{L^2_{\underline{u}}L^2(S_{0,\underline{u}})}+||f(u)\nabla ^i\underline{\beta }||^2_{L^2_{u}L^2(S_{u,0})}) \\ &\qquad \quad +\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _{\underline{H}}\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\!\right)\!\!\left(\!\sum _{i_1+i_2+i_3+i_4\leq 3}\!\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}(K,\check{\sigma })\!\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}K\nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}\psi _H\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}{.} \end{split} \end{equation*}$$

Proof.

Consider the following schematic Bianchi equations:

$$\begin{equation*} \begin{split} \nabla _3\check{\sigma }+\mathrm{div\,}^*\underline{\beta }=\,&\psi _{\underline{H}}\check{\sigma }+\sum _{i_1+i_2+i_3= 1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _{\underline{H}},\\ \nabla _3 K-\mathrm{div\,}\underline{\beta }=\,&\psi _{\underline{H}}K+\sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi \nabla ^{i_3}\psi _{\underline{H}},\\ \nabla _4\underline{\beta }-\nabla K -^*\nabla \check{\sigma }=\,& \psi (K,\check{\sigma })+\sum _{i_1+i_2+i_3=1}\psi ^{i_1}\nabla ^{i_2}\psi _H\nabla ^{i_3}\psi _{\underline{H}}. \end{split} \end{equation*}$$

Commute the first equation with $i$ angular derivatives for $i\leq 3$. We get the equation for $\nabla _3\nabla ^i\check{\sigma }$,

$$\begin{equation} \begin{split} &\nabla _3\nabla ^i\check{\sigma }+\mathrm{div\,}^*\nabla ^i\underline{\beta }\\ &\qquad =\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}{(K,\check{\sigma })}\\ &\qquad \quad +\sum _{i_1+i_2+i_3+i_4=i+1}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _{\underline{H}}. \end{split} {\tag{51}} \end{equation}$$

Notice that in the above equation, there are terms arising from the commutator $[\nabla ^i,\mathrm{div}]\underline{\beta }$, which can be expressed in terms of the Gauss curvature. After substituting also the Codazzi equations for $\underline{\beta }$, we get that these terms have the form of the first term in the above expression. The equation for $\nabla _3\nabla ^i K$ has a similar structure:

$$\begin{equation} \begin{split} &\nabla _3\nabla ^i K-\mathrm{div\,}\nabla ^i\underline{\beta }\\ &\qquad =\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}{(K,\check{\sigma })}\\ &\qquad \quad +\sum _{i_1+i_2+i_3+i_4=i+1}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _{\underline{H}}. \end{split} {\tag{52}} \end{equation}$$

Finally, we have the following structure for $\nabla _4\nabla ^i\underline{\beta }$:

$$\begin{equation} \begin{split} &\nabla _4\nabla ^i\underline{\beta }-\nabla \nabla ^iK -^*\nabla \nabla ^i\check{\sigma }\\ &\qquad = \sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })+\sum _{i_1+i_2+i_3=i-1}\psi ^{i_1}\nabla ^{i_2}K\nabla ^{i_3}(K,\check{\sigma })\\ &\qquad \quad +\sum _{i_1+i_2+i_3+i_4=i+1}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}. \end{split} {\tag{53}} \end{equation}$$

As a shorthand, we denote by $F_{i,1}$ the terms of the form

$$\begin{equation*} \begin{split} F_{i,1}:=&\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}(K,\check{\sigma })+\sum _{i_1+i_2+i_3+i_4= i+1}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _{\underline{H}} \end{split} \end{equation*}$$

and by $F_{i,2}$ the terms of the form

$$\begin{equation*} \begin{split} F_{i,2}&:=\sum _{i_1+i_2+i_3+i_4=i}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })+\sum _{i_1+i_2+i_3= i-1}\psi ^{i_1}\nabla ^{i_2}K\nabla ^{i_3}(K,\check{\sigma })\\ &\quad +\sum _{i_1+i_2+i_3+i_4= i+1}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}. \end{split} \end{equation*}$$

Contracting 53 with $\nabla ^i\underline{\beta }$, integrating in the region $D_{u,\underline{u}}$, applying Proposition 27 and using equations 51 and 52 yield the following identity on the derivatives of the curvature:

$$\begin{equation} \begin{split} &\int _{D_{u,\underline{u}}} f(u)^2 \langle \nabla ^i\underline{\beta },\nabla _4\nabla ^i\underline{\beta }\rangle _\gamma \\ &\qquad =\int _{D_{u,\underline{u}}} f(u)^2 \langle \underline{\beta },\nabla \nabla ^i K+^*\nabla \nabla ^i\sigma \rangle _\gamma +f(u)^2 \langle \nabla ^i\underline{\beta },F_{i,2}\rangle _\gamma \\ &\qquad =\int _{D_{u,\underline{u}}} \!\!-f(u)^2 \langle \mathrm{div\,}\nabla ^i\underline{\beta },\nabla ^iK\rangle _\gamma + f(u)^2\langle \mathrm{div\,}^*\nabla ^i\underline{\beta },\nabla ^i\check{\sigma }\rangle _\gamma +f(u)^2\langle \nabla ^i\underline{\beta },F_{i,2}\rangle _\gamma \\ &\qquad =\int _{D_{u,\underline{u}}} -f(u)^2 \langle \nabla _3\nabla ^iK,\nabla ^iK\rangle _\gamma -f(u)^2\langle \nabla _3\nabla ^i\check{\sigma },\nabla ^i\check{\sigma }\rangle _\gamma \\ &\qquad \quad +\int _{D_{u,\underline{u}}} f(u)^2 \langle \nabla ^i\underline{\beta },F_{i,2}\rangle _\gamma +f(u)^2 \langle \nabla ^i(K,\check{\sigma }),F_{i,1}\rangle _\gamma . \end{split} {\tag{54}} \end{equation}$$

Using Proposition 26, since $\nabla _4 f(u)=0$, we have

$$\begin{equation} \begin{split} &\int f(u)^2 \langle \nabla ^i\underline{\beta },\nabla _4\nabla ^i\underline{\beta }\rangle _\gamma \\ &\qquad =\frac{1}{2} \left(\int _{\underline{H}_{\underline{u}}}f(u)^2|\nabla ^i\underline{\beta }|^2 - \int _{\underline{H}_0}f(u)^2|\nabla ^i\underline{\beta }|^2\right)\\ &\qquad \quad +\int _{D_{u,\underline{u}}} f(u)^2\left(\omega -\frac{1}{2} \mathrm{tr\,}\chi \right)|\nabla ^i\underline{\beta }|^2. \end{split} {\tag{55}} \end{equation}$$

For the terms with $\nabla _3\nabla ^iK$ and $\nabla _3\nabla ^i\check{\sigma }$, we similarly apply Proposition 26, but noting that there is an extra contribution coming from $\nabla _3f(u)$:

$$\begin{equation} \begin{split} &\int _{D_{u,\underline{u}}} f(u)^2 \langle \nabla ^i K,\nabla _3\nabla ^i K\rangle _\gamma \\ &\qquad =-\int _{D_{u,\underline{u}}} f(u)\nabla _3f(u)|\nabla ^i K|^2\\ &\qquad \quad +\frac{1}{2} \left(\int _{H_{u}}f(u)^2|\nabla ^i K|^2 - \int _{H_0}f(u)^2|\nabla ^i K|^2\right)\\ & \qquad \quad +\int _{D_{u,\underline{u}}} f(u)^2(\underline{\omega }-\frac{1}{2} \mathrm{tr\,}\underline{\chi })|\nabla ^i K|^2. \end{split} {\tag{56}} \end{equation}$$

Similarly,

$$\begin{equation} \begin{split} &\int _{D_{u,\underline{u}}} f(u)^2 \langle \nabla ^i \check{\sigma },\nabla _3\nabla ^i \check{\sigma }\rangle _\gamma \\ &\qquad =-\int _{D_{u,\underline{u}}} f(u)\nabla _3f(u)|\nabla ^i \check{\sigma }|^2\\ &\qquad \quad +\frac{1}{2} \left(\int _{H_{u}}f(u)^2|\nabla ^i \check{\sigma }|^2 - \int _{H_0}f(u)^2|\nabla ^i \check{\sigma }|^2\right)\\ & \qquad \quad +\int _{D_{u,\underline{u}}} f(u)^2\left(\underline{\omega }-\frac{1}{2} \mathrm{tr\,}\underline{\chi }\right)|\nabla ^i \check{\sigma }|^2. \end{split} {\tag{57}} \end{equation}$$

Combining 54–57, we thus have the identity

$$\begin{equation*} \begin{split} &\int _{\underline{H}_{\underline{u}}} f(u)^2|\nabla ^i\underline{\beta }|^2+\int _{H_{u}} f(u)^2|\nabla ^i K|^2+\int _{H_{u}} f(u)^2|\nabla ^i \check{\sigma }|\\ &\qquad -2\int _{D_{u,\underline{u}}} f(u)\nabla _3f(u)|\nabla ^i K|^2-2\int _{D_{u,\underline{u}}} f(u)\nabla _3f(u)|\nabla ^i \check{\sigma }|^2\\ &\ = \int _{\underline{H}_{\underline{u}'}} f(u)^2|\nabla ^i\beta |^2+\int _{H_{u'}} f(u)^2|\nabla ^i K|^2+\int _{H_{u'}} f(u)^2|\nabla ^i \check{\sigma }|^2\\ &\quad \quad -2\!\int _{D_{u,\underline{u}}} \! f(u)^2\left(\!\omega \!-\!\frac{1}{2} \mathrm{tr\,}\chi \!\right)|\nabla ^i\underline{\beta }|^2-2\!\int _{D_{u,\underline{u}}} \!f(u)^2\!\left(\underline{\omega }\!-\!\frac{1}{2} \mathrm{tr\,}\underline{\chi }\right)\!(|\nabla ^iK|^2\!+\!|\nabla ^i \check{\sigma }|^2)\\ &\quad \quad +\int _{D_{u,\underline{u}}}f(u)^2\langle \nabla ^i\underline{\beta },F_{i,2}\rangle _\gamma +\int _{D_{u,\underline{u}}}f(u)^2\langle \nabla ^i(K,\check{\sigma }),F_{i,1}\rangle _\gamma . \end{split} \end{equation*}$$

The terms

$$-2\int _{D_{u,\underline{u}}} f(u)\nabla _3f(u)|\nabla ^i K|^2-2\int _{D_{u,\underline{u}}} f(u)\nabla _3f(u)|\nabla ^i \check{\sigma }|^2$$

on the left-hand side, which cannot be controlled⁠²⁸ by the curvature flux (i.e., the integrals of $\nabla ^i$ of the curvature components along $H_u$ or $\underline{H}_{\underline{u}}$), have a favorable sign! This is because the weight function $f$ satisfies $f(u)\nabla _3f(u) <0$. Therefore, we get an inequality for every $i$:

²⁸

In fact, if we do not drop this term, we can control the spacetime integral

$$\int _{D_{u,\underline{u}}} (-f(u)\nabla _3f(u))|\nabla ^i (K,\check{\sigma })|^2,$$

where the weight $(-f(u)\nabla _3f(u))$ can be singular. For weights such as $f(u)=(u-u_*)^{\alpha }$ for $\alpha <\frac{1}{2}$ or $f(u)=(u-u_*)^{\frac{1}{2}}\log ^\beta (\frac{1}{u-u_*})$ for $\beta >\frac{1}{2}$, this bound is logarithmically stronger than simply taking the bound for $\int _{H_u}f(u)^2|\nabla ^i (K,\check{\sigma })|^2$ and integrating in $u$.

✖

$$\begin{equation*} \begin{split} &\int _{\underline{H}_{\underline{u}}} f(u)^2|\nabla ^i\underline{\beta }|^2+\int _{H_{u}} f(u)^2|\nabla ^i K|^2+\int _{H_{u}} f(u)^2|\nabla ^i \check{\sigma }|^2 \\ &\qquad \leq \int _{\underline{H}_{\underline{u}'}} f(u)^2|\nabla ^i\beta |^2+\int _{H_{u'}} f(u)^2|\nabla ^i K|^2+\int _{H_{u'}} f(u)^2|\nabla ^i \check{\sigma }|^2\\ &\qquad \quad +C\left\| f(u)^2\left(\omega -\frac{1}{2} \mathrm{tr\,}\chi \right)\nabla ^i\underline{\beta }\nabla ^i\underline{\beta }\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +C\left\| f(u)^2\left(\underline{\omega }-\frac{1}{2} \mathrm{tr\,}\underline{\chi }\right)\nabla ^i(K,\check{\sigma })\nabla ^i(K,\check{\sigma })\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +C||f(u)^2 \nabla ^i\underline{\beta }F_{i,2}||_{L^1_uL^1_{\underline{u}}L^1(S)}+C||f(u)^2\nabla ^i(K,\check{\sigma }) F_{i,1}||_{L^1_uL^1_{\underline{u}}L^1(S)}. \end{split} \end{equation*}$$

We now add the above inequalities for $i\leq 3$. One can easily check that the terms

$$\sum _{i\leq 3}\left\| f(u)^2\left(\underline{\omega }-\frac{1}{2} \mathrm{tr\,}\underline{\chi }\right)\nabla ^i(K,\check{\sigma })\nabla ^i(K,\check{\sigma })\right\|_{L^1_uL^1_{\underline{u}}L^1(S)},$$

$$\sum _{i\leq 3}||f(u)^2 \nabla ^i\underline{\beta }F_{i,2}||_{L^1_uL^1_{\underline{u}}L^1(S),}$$ and

$$\sum _{i\leq 3}||f(u)^2\nabla ^i(K,\check{\sigma }) F_{i,1}||_{L^1_uL^1_{\underline{u}}L^1(S)}$$

have the form of one of the terms in the statement of the proposition. After applying the Codazzi equation

$$\underline{\beta }=\nabla \psi _{\underline{H}}+\psi (\psi +\psi _{\underline{H}})$$

to one of the $\underline{\beta }$’s, we note that the term

$$\sum _{i\leq 3}\left\| f(u)^2\left(\omega -\frac{1}{2} \mathrm{tr\,}\chi \right)\nabla ^i\underline{\beta }\nabla ^i\underline{\beta }\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}$$

is also one of the terms in the statement of the proposition.

■

To close the energy estimates, we also need to control $\nabla ^i\beta$ in $L^2(H)$ and $\nabla ^i(K,\check{\sigma })$ in $L^2(\underline{H})$. It is not difficult to see, by virtue of the structure of the Einstein equations, that Proposition 28 also holds when all the barred and unbarred quantities are exchanged. The proof is exactly analogous to that of Proposition 28.

Proposition 29.

The following $L^2$ estimates for the curvature components hold:

$$\begin{equation*} \begin{split} &\sum _{i\leq 3}\left(||f(\underline{u})\nabla ^i(K,\check{\sigma })||^2_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}+||f(\underline{u})\nabla ^i\beta ||^2_{L^\infty _{u} L^2_{\underline{u}}L^2(S)}\right) \\ &\qquad \leq \sum _{i\leq 3}\left(||f(\underline{u})\nabla ^i(K,\check{\sigma })||^2_{L^2_{u}L^2(S_{u,0})}+||f(\underline{u})\nabla ^i\beta ||^2_{L^2_{\underline{u}}L^2(S_{0,\underline{u}})}\right) \\ &\qquad \quad +\left\|f(\underline{u})^2\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}\psi _H\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(\underline{u})^2\!\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\!\right)\!\!\left(\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}(K,\check{\sigma })\!\right)\!\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(\underline{u})^2\left(\sum _{i\leq 3}\nabla ^i\beta \right)\left(\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(\underline{u})^2\left(\sum _{i\leq 3}\nabla ^i\beta \right)\left(\sum _{i_1+i_2+i_3+i_4\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}K\nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \quad +\left\|f(\underline{u})^2\left(\sum _{i\leq 3}\nabla ^i\beta \right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}\psi _H\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}{.} \end{split} \end{equation*}$$

We now show that we can control all the nonlinear error terms in the energy estimates. We show this for $K$ and $\check{\sigma }$ in $L^2(H_u)$ and $\underline{\beta }$ in $L^2(\underline{H}_{\underline{u}})$. The other case can be dealt with in a similar fashion (see Proposition 31).

Proposition 30.

There exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ,\Delta _3)$ sufficiently small such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 3}(||f(u)\nabla ^i(K,\check{\sigma })||_{L^\infty _u L^2_{\underline{u}}L^2(S)}+||f(u)\nabla ^i\underline{\beta }||_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}) \leq C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ).$$

Proof.

To prove the curvature estimates, we use Proposition 28. By assumptions of Theorem 2 (see also Remark 11), the two terms corresponding to the initial data are bounded by a constant $C(\mathcal{R}_{\mathrm{ini}} )$ depending only on initial data. Therefore, we need to control the remaining five error terms in Proposition 28. We first look at the term

$$\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}\psi \right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}.$$

Using Propositions 15 and 25, together with the bootstrap assumption (A3), we have

$$\begin{equation*} \begin{split} &\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}\psi \right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\quad \leq C \left(\sum _{i\leq 3}||f(u)\nabla ^i(K,\check{\sigma })||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\right)\left(\sum _{i_1\leq 3}\sum _{i_2\leq 4}||\nabla ^{i_1}\psi ||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}^{i_2}\right)\\ &\qquad \times \left(\sum _{i_3\leq 4}||f(u)\nabla ^{i_3}\psi ||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\right)\left(\sum _{i_4\leq 3}||f(u)\nabla ^{i_4}\psi _{\underline{H}}||_{L^2_uL^\infty _{\underline{u}}L^2(S)}\right)||f(u)^{-1}||_{L^2_u}\\ &\quad \quad +C\epsilon \left(\sum _{i\leq 2}||f(u)\nabla ^i(K,\check{\sigma })||_{L^\infty _uL^2_{\underline{u}}L^2(S)}\!\!\right)||\psi ||_{L^\infty _u L^\infty _{\underline{u}}L^\infty (S)}||f(u)\nabla ^4\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2_uL^2(S)}\\ &\quad \leq C(\mathcal{O}_{\mathrm{ini}} )\Delta _3(1+\Delta _3)\epsilon . \end{split} \end{equation*}$$

The term

$$\begin{equation*} \begin{split} &||f(u)^2\left(\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}(K,\check{\sigma })\right)||_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )\Delta _3(1+\Delta _3)\epsilon \end{split} \end{equation*}$$

similarly as in the previous estimate since by Propositions 16 and 17, $\nabla ^i(K,\check{\sigma })$ satisfies exactly the same estimates as $\nabla ^{i+1}\psi$. We then consider the third nonlinear term

$$\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}.$$

Using Propositions 15 and 25 and the bootstrap assumptions (A3), we have

$$\begin{equation*} \begin{split} &\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 3}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi \nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \leq C\epsilon \left(\sum _{i\leq 3}||f(u)\nabla ^i\underline{\beta }||_{L^\infty _{\underline{u}}L^2_uL^2(S)}\right)\left(\sum _{i_1\leq 3,1\leq i_2\leq 3}||\nabla ^{i_1}\psi ||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}^{i_2}\right)\\ &\qquad \qquad \times \left\|f(u)\sum _{i\leq 3}\nabla ^i(K,\check{\sigma })\right\|_{L^\infty _uL^2_{\underline{u}}L^2(S)}\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )\Delta _3(1+\Delta _3)\epsilon . \end{split} \end{equation*}$$

The fourth nonlinear term can be estimated analogously as the third nonlinear term by

$$\begin{equation*} \begin{split} &\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^i\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 2}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}K\nabla ^{i_4}(K,\check{\sigma })\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} )\Delta _3(1+\Delta _3)\epsilon . \end{split} \end{equation*}$$

As before, this is because by Propositions 16 and 17, $\nabla ^i(K,\check{\sigma })$ satisfies exactly the same estimates as $\nabla ^{i+1}\psi$. Thus it remains to control

$$\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^{i}\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)}.$$

This term can be bounded as follows:

$$\begin{equation*} \begin{split} &\left\|f(u)^2\left(\sum _{i\leq 3}\nabla ^{i}\underline{\beta }\right)\left(\sum _{i_1+i_2+i_3+i_4\leq 4}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}\right)\right\|_{L^1_uL^1_{\underline{u}}L^1(S)} \\ &\ \leq C\left(\sum _{i\leq 3}||f(u)\nabla ^{i}\underline{\beta }||_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}\right)\left(\sum _{i_1\leq 3}\sum _{i_2\leq 3}||\nabla ^{i_1}\psi ||_{L^\infty _uL^\infty _{\underline{u}}L^2(S)}^{i_2}\right)\\ &\quad \times \left(\sum _{i_3\leq 3}||f(\underline{u})\nabla ^{i_3}\psi _H||_{L^2_{\underline{u}}L^\infty _{u}L^2(S)}\right)\left(\sum _{i_4\leq 4}||f(u)\nabla ^{i_4}\psi _{\underline{H}}||_{L^\infty _{\underline{u}}L^2_{u}L^2(S)}\right)||f(\underline{u})^{-1}||_{L^2_{\underline{u}}}\\ &\quad +C\left(\sum _{i\leq 2}||f(u)\nabla ^{i}\underline{\beta }||_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}\right)||f(\underline{u})^{-1}||_{L^2_{\underline{u}}}||f(\underline{u})\nabla ^4\psi _H||_{L^\infty _u L^2_{\underline{u}}L^\infty (S)}\\ &\quad \ \ \times ||f(u)\psi _{\underline{H}}||_{L^2_{u}L^\infty _{\underline{u}}L^\infty (S)}\\ &\ \leq C(\mathcal{O}_{\mathrm{ini}} )\Delta _3(1+\Delta _3)\epsilon . \end{split} \end{equation*}$$

Therefore, gathering all the above estimates, we have

$$\begin{equation*} \begin{split} &\sum _{i\leq 3}\left(||f(u)\nabla ^i(K,\check{\sigma })||_{L^\infty _u L^2_{\underline{u}}L^2(S)}^2+||f(u)\nabla ^i\underline{\beta }||_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}^2\right)\\ &\qquad \leq C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} )+C(\mathcal{O}_{\mathrm{ini}} )\Delta _3(1+\Delta _3)\epsilon , \end{split} \end{equation*}$$

which implies the conclusion of the proposition after taking $\epsilon$ to be sufficiently small.

■

Notice that the schematic equations are symmetric under the change $\nabla _3\leftrightarrow \nabla _4$, $u\leftrightarrow \underline{u},$ and $\psi _H\leftrightarrow \psi _{\underline{H}}$. Since the conditions for the initial data are also symmetric, we also have the following analogous energy estimates for $\nabla ^i\beta$ on $H_u$ and $\nabla ^i(K,\check{\sigma })$ on $\underline{H}_{\underline{u}}$:

Proposition 31.

There exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ,\Delta _3)$ sufficiently small such that whenever $\epsilon \leq \epsilon _0$,

$$\sum _{i\leq 3}(||f(\underline{u})\nabla ^i\beta ||_{L^\infty _u L^2_{\underline{u}}L^2(S)}+||f(\underline{u})\nabla ^i(K,\check{\sigma })||_{L^\infty _{\underline{u}} L^2_{u}L^2(S)}) \leq C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ).{}$$

Propositions 30 and 31 together imply

Proposition 32.

There exists $\epsilon _0=\epsilon _0(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} )$ such that whenever $\epsilon \leq \epsilon _0$,

$$\mathcal{R}\leq C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ).$$

Proof.

Let

$$\Delta _3 \gg C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} ),$$

where $C(\mathcal{O}_{\mathrm{ini}} ,\mathcal{R}_{\mathrm{ini}} )$ is taken to be the maximum of the upper bounds in Propositions 30 and 31. Hence, the choice of $\Delta _3$ depends only on $\mathcal{O}_{\mathrm{ini}}$ and $\mathcal{R}_{\mathrm{ini}}$. Thus, by Propositions 30 and 31, the bootstrap assumption (A3) can be improved by choosing $\epsilon$ sufficiently small depending on $\mathcal{O}_{\mathrm{ini}}$ and $\mathcal{R}_{\mathrm{ini}}$.

■

Combining Propositions 15, 25, and 32, we conclude the proof of Theorem 5. As mentioned previously, standard methods then imply Theorem 2.

9. Nature of the singular boundary

As described by Theorems 3 and 4, we will also prove the regularity and singularity of the boundary $H_{u_*}$ and $\underline{H}_{\underline{u}_*}$. We first prove the regularity of the boundary asserted in Theorem 3.

Proof of Theorem 3.

The fact that $(\mathcal{M}, g)$ can be extended continuously up to and beyond $H_{u_*}$ and $\underline{H}_{\underline{u}_*}$ simply follows from the continuity of the metric components $\Omega$, $\gamma ,$ and $b$ proved in Propositions 1–4. To obtain the higher regularity for $\gamma$, we recall the equations 32, 36, and 37:

$$\begin{eqnarray*} \frac{\partial }{\partial \underline{u}}\Omega ^{-1}=2\omega ,\qquad {\frac{\partial }{\partial \underline{u}}\gamma _{AB}=2\Omega \chi _{AB}},\qquad \frac{\partial }{\partial \underline{u}}b^A=-4\Omega ^2\zeta ^A. \end{eqnarray*}$$

Commuting these equations with $(\frac{\partial }{\partial \theta })^i$ and using the bounds⁠²⁹ for the Ricci coefficients obtained in the proof of Theorem 5, we conclude that

²⁹

Notice that by controlling $\gamma$ and its coordinate angular derivatives $(\frac{\partial }{\partial \theta })^i\gamma$, we can show also that $\frac{\partial }{\partial \theta }$ and $\nabla$ are comparable up to lower-order terms, which allows us to apply the estimates for $\nabla ^i\mathrm{tr\,}\chi$, $\nabla ^i\hat{\chi }$, $\nabla ^i\eta ,$ and $\nabla ^i\underline{\eta }$ to bound the coordinate angular derivatives of the metric components.

✖

$$\sum _{i_1+i_2\leq 4} \sup _{0\leq u\leq {u_*}}\sup _{0\leq \underline{u}\leq \underline{u}_*}\left\|\left(\frac{\partial }{\partial \theta ^1}\right)^{i_1}\left(\frac{\partial }{\partial \theta ^2}\right)^{i_2} (\gamma ,b,\Omega )\right\|_{L^2(U_i(u,\underline{u}))}{\leq C}.$$

The boundedness of $\psi$ and its angular derivatives

$$\sum _{i\leq 3}\|\nabla ^i\psi \|_{L^\infty _u L^\infty _{\underline{u}} L^2(S)}\leq C$$

are already proved in Theorem 5. To control $\psi _{\underline{H}}$ and its angular derivatives on the singular boundary $\underline{H}_{\underline{u}_*}$, we first note that by the smoothness assumption on the interior of the initial hypersurface $\underline{H}_0$, we have that for every fixed $U\in [0,u_*)$,

$${\sum _{i\leq 5}}\sup _{0\leq u\leq U}\|\nabla ^i\psi _{\underline{H}}\|_{L^2(S_{u,0})}\leq C_{{U}}$$

for some finite $C_{U}$. We now revisit the proof of Proposition 13 to bound $\nabla ^i\psi _{\underline{H}}$ up to $i\leq 3$ for $u\in [0,U]$. Restricting to $[0,U]$, $f(u)^{-1}$ is bounded. Therefore, the estimates in 43, 44, and 45 are bounded uniformly in $u$. Finally, 46 can be replaced by the estimate

$$\begin{equation*} \begin{split} &\int _0^{\underline{u}}\sum _{i_1+i_2+i_3+i_4\leq 3}{\|}\nabla ^{i_1}\psi ^{i_2}\nabla ^{i_3}\psi _H\nabla ^{i_4}\psi _{\underline{H}}||_{L^2(S_{u,\underline{u}'})}d\underline{u}' \\ &\qquad \leq C\int _0^{\underline{u}}\left(\sum _{i_1\leq 3}||\nabla ^{i_1}\psi _H||_{L^2(S_{u,\underline{u}'})}\right)\left(\sum _{i_2\leq 3}\sup _{0\leq \underline{u}''\leq \underline{u}}||\nabla ^{i_2}\psi _{\underline{H}}||_{L^2(S_{u,\underline{u}''})}\right)d\underline{u}'. \end{split} \end{equation*}$$

Putting these bounds together, we have

$$\begin{equation*} \begin{split} &\sum _{i\leq 3}\sup _{\substack{0\leq u\leq U\\0\leq \underline{u}'\leq \underline{u}}}\|\nabla ^i\psi _{\underline{H}}\|_{L^2(S_{u,\underline{u}'})}\\ &\qquad \leq C_{{U}}+C\int _0^{\underline{u}}\left(\sum _{i_1\leq 3}||\nabla ^{i_1}\psi _H||_{L^2(S_{u,\underline{u}'})}\!\right)\!\left(\sum _{i_2\leq 3}\sup _{\substack{0\leq u'\leq U\\0\leq \underline{u}''\leq {\underline{u}'}}}||\nabla ^{i_2}\psi _{\underline{H}}||_{L^2(S_{u,\underline{u}''})}\!\right)d\underline{u}', \end{split} \end{equation*}$$

which implies

$$\begin{equation} \sum _{i\leq 3}\sup _{\substack{0\leq u\leq U\\0\leq \underline{u}\leq \underline{u}_*}}\|\nabla ^i\psi _{\underline{H}}\|_{L^2(S_{u,\underline{u}})}\leq {C_U} {\tag{58}} \end{equation}$$

after applying Gronwall’s inequality.

To conclude the proof, it remains to control $\nabla _3\nabla ^i\psi$ and $\nabla _3\nabla ^i\psi _{\underline{H}}$ for $i\leq 2$. Since $\underline{\eta }$ obeys a $\nabla _3$ equation (see 9), by directly controlling the right-hand side of the null structure equation (commuted with angular derivatives) and using the bounds in Theorem 5, we get

$$\sum _{i\leq 2}{\sup _{\substack{0\leq u\leq U\\0\leq \underline{u}\leq \underline{u}_*}}}\|\nabla _3\nabla ^i\underline{\eta }\|_{L^2(S_{u,\underline{u}})}\leq C_U.$$

To control the term $\nabla _3\nabla ^i\eta$, notice that combining the $\nabla _3\underline{\eta }$ equation in 9 and the equations in 7, we have

$$\nabla _3\eta =-\nabla _3\underline{\eta }+2\nabla _3\nabla (\log \Omega )=\underline{\chi }\cdot (\underline{\eta }-\eta )-\underline{\beta }-\nabla \underline{\omega }-4\underline{\omega }\nabla (\log \Omega )-2\underline{\chi }\cdot \nabla (\log \Omega ).$$

Upon expressing $\underline{\beta }$ in terms of $\psi _{\underline{H}}$ using the Codazzi equation in 10, commuting the equation with $\nabla ^i,$ and using the bound 58, we get

$$\begin{equation} \sum _{i\leq 2}{\sup _{\substack{0\leq u\leq U\\0\leq \underline{u}\leq \underline{u}_*}}}\|\nabla _3\nabla ^i\eta \|_{L^2(S_{u,\underline{u}})}\leq C_{{U}}. {\tag{59}} \end{equation}$$

Finally, we control the terms $\nabla _3\nabla ^i\psi _{\underline{H}}$. Commuting the null structure equations for $\nabla _4\psi _{\underline{H}}$ in 8 and 9 with $\nabla _3\nabla ^i$, we have

$$\begin{equation*} \begin{split} &\nabla _4\nabla _3\nabla ^i\psi _{\underline{H}}\\ &\qquad =\sum _{\substack{j_1+j_2+j_3+j_4=1\\i_1+i_2+i_3+i_4=i}}(\nabla ^{i_1}\psi _{\underline{H}}^{j_1}\nabla _3^{j_2}\nabla ^{i_2}\psi ^{i_3}\nabla _3^{j_3}\nabla ^{i_4}K+\nabla ^{i_1}\psi _{\underline{H}}^{j_1}\nabla _3^{j_2}\nabla ^{i_2}\psi ^{i_3}\nabla _3^{j_3}\nabla ^{i_4}\nabla \psi )\\ &\qquad \quad +\sum _{\substack{j_1+j_2+j_3=1\\i_1+i_2+i_3+i_4+i_5=i}}\nabla ^{i_1}\psi _{\underline{H}}^{j_1}\nabla _3^{j_2}\nabla ^{i_2}\psi ^{i_3}\nabla _3^{j_3}\nabla ^{i_4}\psi \nabla ^{i_5}\psi \\ &\qquad \quad +\sum _{\substack{j_1+j_2+j_3+j_4=1\\i_1+i_2+i_3+i_4+i_5=i}}\nabla ^{i_1}\psi _{\underline{H}}^{j_1}\nabla _3^{j_2}\nabla ^{i_2}\psi ^{i_3}\nabla _3^{j_3}\nabla ^{i_4}\psi _{\underline{H}}\nabla _3^{j_4}\nabla ^{i_5}\psi _H. \end{split} \end{equation*}$$

Estimating directly the right-hand side of the null structure equations or the Bianchi equations, we can easily show that

$$\sum _{i\leq 2}{\sup _{0\leq u\leq U}}\|\nabla _3(\nabla ^iK,\nabla ^i\underline{\eta },\nabla ^i\psi _H)\|_{L^1_{\underline{u}}L^2(S)}\leq C_{{U}}.$$

Using also 59, we thus have

$$\begin{equation*} \begin{split} &\sum _{i\leq 2}{\sup _{0\leq u\leq U}}\|\nabla _3\nabla ^i\psi _{\underline{H}}\|_{L^2(S_{u,\underline{u}})}\\ &\qquad \leq C_{{U}}+C_{{U}}\int _0^{\underline{u}}\sum _{i_1+i_2+i_3+i_4\leq 2}\|\nabla ^{i_1}\psi ^{i_2}\nabla _3\nabla ^{i_3}\psi _{\underline{H}}\nabla ^{i_4}\psi _H\|_{L^2(S_{u,\underline{u}'})} d\underline{u}'. \end{split} \end{equation*}$$

Using Gronwall’s inequality, we get

$$\sum _{i\leq 2}{\sup _{\substack{0\leq u\leq U\\0\leq \underline{u}\leq \underline{u}_*}}}\|\nabla _3\nabla ^i\psi _{\underline{H}}\|_{L^2(S_{u,\underline{u}})}\leq C_{{U}}.$$

In particular combining the above estimates, we obtain

$$\displaystyle \sum _{\substack{i\leq 3-j,\,j\leq 1}}\sup _{0\leq u\leq U}\|\nabla ^j\nabla ^i(\psi _{\underline{H}},\psi )\|_{L^2(S_{u,\underline{u}_*})}\leq C_{{U}}$$

on $\underline{H}_{\underline{u}_*}$, as desired.

■

Finally, we move to the proof of Theorem 4. First, we prove

Proposition 33.

Suppose, in addition to the assumptions in Theorem 2, $\hat{\chi }$ initially obeys

$$\int _0^{\underline{u}_*} |\hat{\chi }\restriction _{\gamma }(\underline{u}')|^2 d\underline{u}' =\infty ,$$

along an outgoing null generator $\gamma$ of $H_0$. Let $\Phi _u(\gamma )$ be the image of $\gamma$ under the one-parameter family of diffeomorphisms generated by $\underline{L}$. Then

$$\int _0^{\underline{u}_*} (\mathrm{tr\,}\chi \restriction _{\Phi _u(\gamma )}(\underline{u}'))^2+|\hat{\chi }\restriction _{\Phi _u(\gamma )}(\underline{u}')|^2 d\underline{u}' =\infty ,$$

holds for every $0\leq u <u_*$.

Similarly suppose, in addition to the assumptions in Theorem 2, $\hat{\underline{\chi }}$ initially obeys

$$\int _0^{\underline{u}_*} |\hat{\underline{\chi }}\restriction _{\gamma }(u')|^2 du' =\infty ,$$

along an outgoing null generator $\gamma$ of $\underline{H}_0$. Let ${\underline{\Phi }}_{\underline{u}}(\gamma )$ be the image of $\gamma$ under the one-parameter family of diffeomorphisms generated by $L$. Then

$$\int _0^{u_*} (\mathrm{tr\,}\underline{\chi }\restriction _{\Phi _{\underline{u}}(\gamma )}(u'))^2+|\hat{\underline{\chi }}\restriction _{\Phi _{\underline{u}}(\gamma )}(u')|^2 du' =\infty ,$$

holds for every $0\leq \underline{u}<\underline{u}_*$.

Proof.

Fix $U\in (0,u_*{)}$. Suppose

$$\begin{equation} \int _0^{\underline{u}_*} (\mathrm{tr\,}\chi \restriction _{\Phi _U(\gamma )}(\underline{u}'))^2 d\underline{u}' <\infty . {\tag{60}} \end{equation}$$

We want to show that under the assumption 60, we have

$$\int _0^{\underline{u}_*} |\hat{\chi }\restriction _{\Phi _U(\gamma )}(\underline{u}')|^2 d\underline{u}' =\infty {,}$$

which will then imply the desired conclusion.

Using 60, define $h:[0,\underline{u}_*)\to \mathbb{R}$ by

$$h(\underline{u})=|\mathrm{tr\,}\chi \restriction _{\Phi _U(\gamma )}(\underline{u})|$$

such that

$$\int _0^{\underline{u}_*} h(\underline{u}')^2 d\underline{u}'<\infty .$$

Consider the following null structure equation for $\mathrm{tr\,}\chi$:

$$\nabla _3 \mathrm{tr\,}\chi + \mathrm{tr\,}\underline{\chi }\mathrm{tr\,}\chi =2\underline{\omega }\mathrm{tr\,}\chi -2K+2\mathrm{div\,}\eta +2|\eta |^2.$$

Along the integral curve of $-e_3$ emanating from $\Phi _u(\gamma )$, we thus have

$$\begin{equation*} \begin{split} &\frac{d}{du}\left(e^{\int _U^u(\Omega \mathrm{tr\,}\underline{\chi }-2\Omega {\underline{\omega }})\restriction _{\Phi _{u'}(\gamma )}(\underline{u})du' }\mathrm{tr\,}\chi \restriction _{\Phi _{u}(\gamma )}(u)\right)\\ &\qquad =e^{\int _U^u(\Omega \mathrm{tr\,}\underline{\chi }-2\Omega {\underline{\omega }})\restriction _{\Phi _{u'}(\gamma )}(\underline{u})du'}(-2K + 2\mathrm{div\,}\eta +2|n|^2). \end{split} \end{equation*}$$

By the estimates derived in the proof of Theorem 5, $K$, $\nabla \eta$, $\eta$ are bounded and $\mathrm{tr\,}\underline{\chi }$, $\underline{\omega }$ are in $L^1_u L^\infty (S)$. Therefore,

$$\begin{equation} |\mathrm{tr\,}\chi \restriction _{\Phi _u(\gamma )}(\underline{u})|\leq Ch(\underline{u})\quad \text{for all }u. {\tag{61}} \end{equation}$$

Consider the following null structure equation for $\hat{\chi }$:

$$\nabla _3\hat{\chi }+\frac{1}{2} \mathrm{tr\,}\underline{\chi }\hat{\chi }=\nabla \widehat{\otimes } \eta +2\underline{\omega }\hat{\chi }-\frac{1}{2} \mathrm{tr\,}\chi \hat{\underline{\chi }}+\eta \widehat{\otimes } \eta .$$

Contract this equation with $\hat{\chi }$ to get

$$\frac{1}{2} \nabla _3|\hat{\chi }|^2+\frac{1}{2} \mathrm{tr\,}\underline{\chi }|\hat{\chi }|^2 -2\underline{\omega }|\hat{\chi }|^2=\left(\nabla \widehat{\otimes } \eta -\frac{1}{2} \mathrm{tr\,}\chi \hat{\underline{\chi }}+\eta \widehat{\otimes } \eta \right)\cdot \hat{\chi },$$

which implies

$$\left|\nabla _3|\hat{\chi }|+\frac{1}{2} \mathrm{tr\,}\underline{\chi }|\hat{\chi }| -2\underline{\omega }|\hat{\chi }|\right|\leq |\nabla \widehat{\otimes } \eta |+\left|\frac{1}{2} \mathrm{tr\,}\chi \hat{\underline{\chi }}\right| +|\eta \widehat{\otimes } \eta |.$$

This implies that along the integral curve of $e_3$, we have

$$\begin{equation*} \begin{split} &\left|\frac{d}{du} (e^{\int ^u_U (\frac{1}{2}{\Omega {\mathrm{tr\,}}\underline{\chi }}-2{\Omega }\underline{\omega })\restriction _{\Phi _{u'}(\gamma )} (\underline{u}) du'} |\hat{\chi }{|} \restriction _{\Phi _{u}(\gamma )} (\underline{u}))\right|\\ &\qquad \leq 2e^{\int ^u_U (\frac{1}{2}{\Omega {\mathrm{tr\,}}\underline{\chi }}-2{\Omega }\underline{\omega })\restriction _{\Phi _{u'}(\gamma )} (\underline{u}) du'}\left(|\nabla \widehat{\otimes } \eta |+\left|\frac{1}{2} \mathrm{tr\,}\chi \hat{\underline{\chi }}\right| +|\eta \widehat{\otimes } \eta |\right). \end{split} \end{equation*}$$

Using again the fact that $K$, $\nabla \eta$, $\eta$, $\mathrm{tr\,}\underline{\chi }$, $\hat{\underline{\chi }}$, $\underline{\omega }$ are bounded for $u\leq U$, as well as the estimate 61, we have

$$\begin{equation*} \begin{split} &\left|\left(e^{\int ^u_U (\frac{1}{2}{\Omega {\mathrm{tr\,}}\underline{\chi }}-2{\Omega }\underline{\omega })\restriction _{\Phi _{u'}(\gamma )} (\underline{u}) du'} |\hat{\chi }{|} \restriction _{\Phi _{u}(\gamma )} (\underline{u})\right)-\left(e^{\int ^u_U ({\frac{1}{2}\Omega {\mathrm{tr\,}}\underline{\chi }}-2{\Omega }\underline{\omega })\restriction _{\gamma } (\underline{u}) du'} |\hat{\chi }{|} \restriction _{\gamma } (\underline{u})\right)\right|\\ &\qquad \leq C_U(1+h(\underline{u})). \end{split} \end{equation*}$$

Notice that $e^{\int ^u_U ({\frac{1}{2}\Omega {\mathrm{tr\,}}\underline{\chi }}-2{\Omega }\underline{\omega })\restriction _{\Phi _{u'}(\gamma )} (\underline{u}) du'}$ is bounded above and below uniformly in $\underline{u}$. Taking the $L^2_{\underline{u}}$ norm implies that for $u\leq U$, we have

$$\begin{equation*} \int _0^{\underline{u}_*}|\hat{\chi }\restriction _{\Phi _u(\gamma )}(\underline{u}')|^2 d\underline{u}'\geq c\int _0^{\underline{u}_*} |\hat{\chi }\restriction _{\gamma }(\underline{u}')|^2 d\underline{u}'-C-C\int _0^{\underline{u}_*} h^2(\underline{u}')d\underline{u}' =\infty \end{equation*}$$

by the assumption of the proposition. The blowup for $\underline{\chi }$ can be proved in a similar manner.

■

This implies

Proposition 34.

Suppose the assumptions of Theorem 4 hold. Then, in a neighborhood of any point on $\underline{H}_{\underline{u}_*}$, $|{\chi }|^2$ is not integrable with respect to the spacetime volume form. Similarly, in a neighborhood of any point on $H_{u_*}$, $|{\underline{\chi }}|^2$ is not integrable with respect to the spacetime volume form.

Proof.

We begin with $|\underline{\chi }|^2$ near $H_{u_*}$. By definition, the image of the initial incoming null generator under the map $\underline{\Phi }_{\underline{u}}$ defined in Proposition 33 has constant $\underline{u}$, $\theta ^1,$ and $\theta ^2$ values. Also, by Propositions 1 and 2, the spacetime volume element $2\Omega ^2\sqrt {\det \gamma }$ is bounded uniformly above and below. Therefore, for any neighborhood $\mathcal{N}$ of $p=(u,\underline{u}_*,\theta ^1,\theta ^2)\in \underline{H}_{\underline{u}_*}$, we have

$$\begin{equation*} \begin{split} &\int _{\mathcal{N}} ((\mathrm{tr\,}\underline{\chi })^2+|\hat{\underline{\chi }}|^2) \\ &\quad \geq c \int _{\theta ^2-\delta }^{{\theta ^2}+\delta }\!\int _{\theta ^1-\delta }^{\theta ^1+\delta }\!\int _{u-\delta }^{u+\delta }\!\int _{\underline{u}_*-\delta }^{\underline{u}_*}((\mathrm{tr\,}\underline{\chi })^2\!+\!|\hat{\underline{\chi }}|^2)(u',\underline{u}',(\theta ^1)',(\theta ^2)') d\underline{u}'\, du'\,d(\theta ^1)'\, d(\theta ^2)'\\ &\quad =\infty , \end{split} \end{equation*}$$

by Proposition 33.

To prove the corresponding statement for $|{\chi }|^2$ near ${\underline{H}_{\underline{u}_*}}$, we first change to the coordinate system $(u,\underline{u},\tilde{\theta }^1(\underline{u};u,\theta ),\tilde{\theta }^2(\underline{u};u,\theta ))$ such that $\underline{L}=\frac{\partial }{\partial u}$. This coordinate system can be constructed by solving the ordinary differential equations

$${\frac{d}{du}\tilde{\theta }^A(u;\underline{u},\theta )=-b^A(u,\underline{u},\tilde{\theta }^1,\tilde{\theta }^2),}$$

with initial condition⁠³⁰

³⁰

We note that since we do not have a global coordinate chart on $S_{0,0}$, the above ODE only makes sense in $(\Phi _u\circ \underline{\Phi }_{\underline{u}})(U_i)\cap (\underline{\Phi }_{\underline{u}}\circ \Phi _u)(U_j)$, where $U_i$, $U_j$ are coordinate charts on $S_{0,0}$ and $\Phi _u$ and $\underline{\Phi }_{\underline{u}}$ are as defined in Proposition 33. Nevertheless, since $\Phi _u\circ \underline{\Phi }_{\underline{u}}$ and $\underline{\Phi }_{\underline{u}}\circ \Phi _u$ are both diffeomorphisms between $S_{0,0}$ and $S_{u,\underline{u}}$, for every point $p\in S_{u,\underline{u}}$, there exists $i$ and $j$ such that $p\in (\Phi _u\circ \underline{\Phi }_{\underline{u}})(U_i)\cap (\underline{\Phi }_{\underline{u}}\circ \Phi _u)(U_j)$, where this change of coordinates makes sense.

✖

$${\tilde{\theta }^A(0;\underline{u},\theta )}=\theta ^A.$$

By 37, as well as the estimates for $\zeta$, $\Omega$ and their derivatives, $b^A$ and the following first derivatives of $b^A$ are uniformly bounded:

$$|b^A|,\,\left|\frac{\partial b^A}{\partial \underline{u}}\right|,\,\left|\frac{\partial b^A}{\partial \theta ^B}\right|\leq C.$$

Therefore,

$$\left|\frac{\partial \tilde{\theta }^A}{\partial u}\right|,\,\left|\frac{\partial \tilde{\theta }^A}{\partial \underline{u}}\right|,\,\left|\frac{\partial \tilde{\theta }^A}{\partial \theta ^B}\right|\leq C.$$

In the new coordinate system, we apply the same argument as in the case for $|{\underline{\chi }}|^2$ near $H_{u_*}$ and have the estimate

$$\int _{\mathcal{N}} \left((\mathrm{tr\,}\chi )^2+|\hat{\chi }|^2\right) =\infty$$

for any neighborhood $\mathcal{N}$ of any point $p\in {\underline{H}_{\underline{u}_*}}$, as desired.

■

Finally, this allows us to conclude that the Christoffel symbols do not belong to $L^2$:

Proposition 35.

Suppose the assumptions of Theorem 4 hold. Then, the Christoffel symbols in the $(u,\underline{u},\theta ^1,\theta ^2)$ coordinate system are not in $L^2$ in a neighborhood of any point on $H_{u_*}$ or $\underline{H}_{\underline{u}_*}$.

Proof.

Recall that the metric in the $(u,\underline{u},\theta ^1,\theta ^2)$ coordinates takes the form

$$g=-2\Omega ^2(du\otimes d\underline{u}+d\underline{u}\otimes du)+\gamma _{AB} (d\theta ^A-b^A du)\otimes (d\theta ^B-b^B du).$$

Note that

$$g^{u\underline{u}}=-\frac{1}{2} \Omega ^{-2},\qquad g^{u\alpha }=0\text{ for }\alpha \neq \underline{u}.$$

One computes that

$$\begin{equation*} \begin{split} \Gamma ^{u}_{AB}=&-\frac{1}{2} g^{u \underline{u}} \frac{\partial }{\partial \underline{u}}g_{AB} =\frac{1}{4 \Omega ^2} \frac{\partial }{\partial \underline{u}}\gamma _{AB} =\frac{1}{2 \Omega } \chi _{AB}. \end{split} \end{equation*}$$

Since $\frac{1}{2}\leq \Omega \leq 2$ and $\gamma$ is uniformly bounded and positive definite, $\Gamma ^u_{AB}$ is not in $L^2$ in a neighborhood of any point on the singular boundary $\underline{H}_{\underline{u}_*}$ in the $(u,\underline{u},\theta ^1,\theta ^2)$ coordinate system.

To show that the incoming hypersurface $H_{u_*}$ is singular, first notice that

$$g^{\underline{u}u}=-\frac{1}{2} \Omega ^{-2},\qquad g^{\underline{u}A}=-\frac{1}{2} \Omega ^{-2}b^A,\qquad g^{\underline{u}\underline{u}}=0.$$

We then compute

$$\begin{equation*} \begin{split} \Gamma ^{\underline{u}}_{AB}=\,&\frac{1}{2} g^{\underline{u}u}\left(\frac{\partial }{\partial \theta ^A}g_{Bu}+\frac{\partial }{\partial \theta ^B}g_{Au} -\frac{\partial }{\partial u}g_{AB}\right)\\ &+\frac{1}{2} g^{\underline{u}C}\left(\frac{\partial }{\partial \theta ^B}g_{AC}+\frac{\partial }{\partial \theta ^A}g_{BC}-\frac{\partial }{\partial \theta ^C}g_{AB}\right)\\ =\,&\frac{1}{4 \Omega ^2} \left(\frac{\partial }{\partial u}\gamma _{AB}-\frac{\partial }{\partial \theta ^B}(\gamma _{AC} b^C)-\frac{\partial }{\partial \theta ^A}(\gamma _{BC} b^C)\right.\\ &\qquad \left.-b^C\left(\frac{\partial }{\partial \theta ^B}\gamma _{AC}+\frac{\partial }{\partial \theta ^A}\gamma _{BC}-\frac{\partial }{\partial \theta ^C}\gamma _{AB} \right)\right)\\ =\,&\frac{1}{2 \Omega } \underline{\chi }_{AB}+\text{regular terms}, \end{split} \end{equation*}$$

where the regular terms denote metric components and their derivatives that are uniformly bounded by the estimates proved in the previous sections. By the same reasoning as in the case near $\underline{H}_{\underline{u}_*}$, $\Gamma ^{\underline{u}}_{AB}$ is not in $L^2$ in a neighborhood of any point on the singular boundary $H_{u_*}$ in the $(u,\underline{u},\theta ^1,\theta ^2)$ coordinate system.

■

This concludes the proof of Theorem 4.

Figure 1.

The Penrose diagram of Reissner–Nordström spacetimes

Figure 2.

Region of existence in Conjecture 1

Figure 3.

Region of existence in Theorem 1

Figure 4.

Domains of existence

Figure 5.

Perturbations in the black hole interior of Dafermos spacetimes

Figure 6.

The basic setup