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
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 solutionFootnote1 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 Reference 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 diagramsFootnote2 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 sharedFootnote3 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.
In a seminal work Dafermos Reference 7Reference 8 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 Reference 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 Reference 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 Reference 7Reference 8Reference 9 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 Reference 7Reference 8Reference 9 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,
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 Reference 8. The instability of the Reissner–Nordström Cauchy horizon is in fact already suggested by a linear analysis (see Reference 4Reference 20Reference 23). For a spherically symmetric solution to the linear wave equation which has a polynomially decaying (in the Eddington–Finkelstein coordinates) tailFootnote5 along the event horizon, there is a singularity in a ($C^0$)-regular coordinate system near the Cauchy horizon of the strengthFootnote6
This statement regarding the linear wave equation can be inferred using the methods in Reference 7 for the nonlinear coupled Einstein–Maxwell–scalar field system.
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 Reference 7Reference 8, 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 Reference 8 also blows up at a rate given by Equation 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 Equation 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 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 Reference 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 Reference 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 Reference 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.Footnote8 Thus, we show that the weak null singularity of Dafermos Reference 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 Reference 23 suggesting that the Reissner–Nordström Cauchy horizon is unstable. This was also confirmed by the numerical work by Simpson and Penrose Reference 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 understoodFootnote9 until the first study of weak null singularity carried out by Hiscock Reference 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 Reference 25Reference 26 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 Reference 1Reference 2Reference 3).
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 Reference 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 Reference 22. While this class of spacetime is more restrictive, as discussed in Reference 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 Reference 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 Reference 24, Khan and Penrose Reference 14, and Szekeres Reference 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 Reference 18Reference 19, 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 Reference 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 areFootnote10 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 Reference 18Reference 19 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 Reference 18 on the interaction of two impulsive gravitational waves. In particular the renormalized energy of Reference 18Reference 19 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 $p>1$.
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 Reference 5Reference 15Reference 16.
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 $\{e_1,e_2,e_3,e_4\}$:
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:
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
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
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:
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.
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:
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, Reference 5Reference 15), the standard approach to obtain a priori bounds is to couple the $L^2$ estimates for the curvature components
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 Reference 18Reference 19). 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 Reference 18Reference 19 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,
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
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.Footnote13
13
The can be compared with the renormalization introduced in Reference 19 and Reference 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.
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 Reference 18Reference 19, 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 Equation 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:
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
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
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.Footnote15 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 Reference 18Reference 19, 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 Reference 18Reference 19, 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 smoothFootnote16 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.
In this paper, we will consider spacetime solutions to the vacuum Einstein equations Equation 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,
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 Reference 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
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 frameFootnote17$\{e_A\}_{A=1,2}$ 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.
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 Reference 15, Chapter 3.1).
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 $S_{u,\underline{u}}$):
$$\left(\mathrm{curl\,}\phi \right)_{A_1\cdots A_r}:={\epsilon } \mkern -7mu /\,^{BC}\nabla _B\phi _{CA_1\cdots A_r}.$$ Define also the trace of totally symmetric tensors to be
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:
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
Notice that we have obtained a system for the renormalized curvature components in which the curvature components $\alpha$ and $\underline{\alpha }$ do not appear.Footnote18
18
Moreover, compared to the renormalization in Reference 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:Footnote19
19
Notice that this definition is different form that in Reference 19, since in the context of the present paper $\mathrm{tr\,}\chi$ and $\mathrm{tr\,}\underline{\chi }$ verify different bounds compared to Reference 19.
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.,
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
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
In order to prove Theorem 2, we will establish a priori estimates for the geometric quantities in the above norms:
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 Reference 5Reference 19 for a proof that the a priori estimates imply the existence theorem.
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 Reference 5, Chapter 2.
Assume for simplicity that $S_{0,0}$ is a standard sphere of radius $1$. IntroduceFootnote21 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.
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
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 satisfyFootnote22
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 $|J|=j_1+j_2$.
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
We will show the estimates separately for $\mathrm{tr\,}\chi$ and $\hat{\chi }$. By Equation 22, Equation 26 holds for $\hat{\chi }$ when $i=0$. To derive this bound for $\mathrm{tr\,}\chi$, notice that by the ODE Equation 23 for $\Phi$, the initial conditions Equation 24, and the bound Equation 22 for $|\hat{\chi }|^2$, we have
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 Equation 21, we thus have
Using this bound and commuting the ODE Equation 23 with $\frac{\partial }{\partial \theta }$, we also have that for up to $N$ coordinate angular derivatives $\frac{\partial }{\partial \theta }$,
Finally, we notice that by Equation 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, Equation 26 follows from Equation 30 and Equation 31.
Estimates for $\nabla ^i K$
To control $\nabla ^i K$, we simply notice that by Equation 29, we have
On $H_0$, since $\Omega =1$,$\eta =\zeta$. Thus combining the transport equation for $\zeta$ in Equation 9 and the Codazzi equation for $\beta$ in Equation 10, and rewriting in ${\mathcal{L}} \mkern -9mu /\,$ (instead of $\nabla _4$), we have
Recall from Equation 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
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, Equation 9 and the Gauss equation in Equation 10 imply that
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
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 (Equation A1):
We then show that we can control $\gamma$ under the bootstrap assumption (Equation A1):
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.
We can also bound $b$ under the bootstrap assumption, thus controlling the full spacetime metric:
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 (Equation 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.
We also have the following bounds for the $p=\infty$ case by integrating along the integral curves of $e_3$ and $e_4$:
5.3. Sobolev embedding
Using the estimates for the metric $\gamma$ in Proposition 2, we have the following Sobolev embedding theorem:
$$(\mathrm{curl\,}\phi )_{A_1\cdots A_r}={\epsilon } \mkern -7mu /\,^{BC}\nabla _B\phi _{CA_1\cdots A_r},$$ where ${\epsilon } \mkern -7mu /\,$ is the volume form associated to the metric $\gamma$. Recall also that the trace is defined to be
For the special case that $\phi$ is a symmetric traceless 2-tensor, we only need to know its divergence:
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 (Equation A1) and will show that the constant in Equation 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 $\psi$.
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 $||f(u)\cdot ||_{L^2_u L^\infty _{\underline{u}}L^\infty (S)}$.
Using instead the equation for $\nabla _3\psi _H$, we obtain the following estimates in a completely analogous manner:
By the Sobolev embedding theorems given by Proposition 7, we have thus closed our bootstrap assumption (Equation 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.
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:
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 Reference 5Reference 15Reference 16. 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 $L^2(S)$.
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.
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.
Once we have the estimates for $\nabla ^4\mathrm{tr\,}\chi$, we can control $\nabla ^4\hat{\chi }$ using elliptic estimates:
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 $\nabla ^4\hat{\chi }$:
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.
A similar proof shows that the conclusion of Proposition 21 holds also for $\nabla ^3\underline{\eta }$:
We now move to the estimates for $\nabla ^4\underline{\omega }$:
By switching $\underline{\omega }$ and $\omega$ as well as $e_3$ and $e_4$, we also have the following estimates for $\nabla ^4\omega$:
We have thus controlled the fourth angular derivatives of all Ricci coefficients and have closed the bootstrap assumption (Equation 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:
8. Estimates for curvature
In this section, we derive and prove the energy estimates. To this end, we introduce the following bootstrap assumptions:
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:
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.
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.
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).
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 $\underline{H}_{\underline{u}}$:
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.
Finally, we move to the proof of Theorem 4. First, we prove
This implies
Finally, this allows us to conclude that the Christoffel symbols do not belong to $L^2$:
The author thanks Mihalis Dafermos for suggesting the problem and sharing many insights from the works Reference 7Reference 8Reference 9 as well as offering valuable comments on an earlier version of the manuscript. He thanks Igor Rodnianski for very helpful suggestions. He also thanks Spyros Alexakis, Amos Ori, and Yakov Shlapentokh-Rothman for stimulating discussions. Finally, he is grateful for the suggestions given by the anonymous referees.
Most of the work was carried out when the author was at Princeton University and University of Pennsylvania.
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