Asymptotic Curvature Decay and Removal of Singularities of Bach-Flat Metrics

We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the proof we analyze the decay rates of solutions to the Bach-flat equation linearized around a flat metric. This classification is used to prove that Bach-flat cones are in fact ALE of order $\tau$ for any $\tau<2$. This result is then used to prove the removal of singularities theorem.


Introduction
In dimension 4, the Euler-Lagrange equations for the functional W(g) := M |W g | 2 dV g (1. 1) where W g is the Weyl curvature, are given by where W ijkl and R kl correspond to the components of the Weyl and Ricci tensors respectively [2], [4]. B ij are the components of the Bach tensor, and a metric which is critical for W is called Bach-flat.
A smooth Riemannian manifold (X 4 , g) is called an asymptotically locally Euclidean end of order τ if there exists a finite subgroup Γ ⊂ SO(4) which acts freely on R 4 \B(0, R) and a C ∞ diffeomorphism φ : X → (R 4 \B(0, R)) /Γ such that using this identification for any partial derivative of order k as r → ∞. We say an end is ALE of order 0 if there exist coordinates as above so that as r → ∞.
Theorem: 1.1. Let (X 4 , g) be a complete, noncompact four-dimensional Bach-flat space with zero scalar curvature which is ALE of order 0. Assume further that Then the cone is ALE of order τ for any τ < 2.
This theorem is an extension of theorem 1.1 of [7], where under the same hypotheses on curvature and Sobolev constant and also assuming b 1 (X) < ∞ where b 1 (X) is the first Betti number it is proved that the end is ALE of order zero. where C s and V 1 are positive constants. Then the metric g extends to B(x, 1) as a smooth orbifold metric. This is an extension of theorem 6.4 of [6], and indeed this result was suggested in the work of Gang Tian and Jeff Viaclovsky [6], [7]. There removal of singularities is proved for certain subclasses of Bachflat manifolds, specifically half-conformally flat metrics, metrics with harmonic curvature, and constant scalar curvature Kähler metrics. The proof uses certain Kato inequalities which are satisfied in these special cases to improve the decay of the Ricci tensor, which in turn can be used to get improved decay of the full curvature tensor. The techniques used herin also suffice to give this result assuming that the metric has harmonic curvature, that is, We now give an outline of the rest of the paper. In section 2 we decompose the action of the Bilaplacian acting on symmetric two-tensors according to the radial separation of variables on a cone. In section 3 we compute the linearization of the Bach-flat condition around a flat metric. We observe that after applying a change of diffeomorphism gauge and possibly a conformal change this equation is equivalent to the equation ∆ 2 h = 0 where h is the perturbation of the metric. Using the separation of variables we further simplify this equation and classify the solutions in terms of their decay or growth rates at infinity. Section 4 gives the proof of theorem 1.1, which consists of considering the Bach-flat equation at infinity in the cone as a perturbation of a flat metric, and using our analysis of the linearized equation. Section 5 uses theorem 1.1 to prove theorem 1.2.
Acknowledgements: The author would like to express deep gratitude to Gang Tian for suggesting this problem and suggesting that the arguments of [1] could be used to prove theorem 1.2.

Decomposition of the Bilaplacian
In this section we will decompose the action of the Bilaplacian on symmetric two-tensors on R 4 according to the cone structure. These tensors naturally decompose according to radial and tangential directions. We reintroduce the notation of [1] and recall the decomposition of the Laplacian proved there. These lemmas are then used in a straightforward way to decompose the action of the Bilaplacian. The calculation is greatly simplified by restricting to R 4 , where we completely understand the curvature tensor, and how it decomposes according to the cone structure.
Write R 4 , g flat as C(S 3 , g can ), the cone over the three-sphere with its canonical metric. Let P : T (1, S 3 ) → T (r, S 3 ) denote the identification of tangent bundles induced by parallel transport along radial geodesics, and let ∇ r denote the Riemannian connection with respect to the induced metric on (r, S 3 ). Define the connection A basic formula is then Also let d r , δ r , tr r be defined with respect to the induced metric on (r, S 3 ) and let d r denote exterior differentiation on (r, S 3 ). Then define The following formulae are immediate Now say e is a vector tangent to (r, S 3 ) and e * is its dual 1-form. Let η be a 1-form such that Some simple formulas are then Now fix ∂ ∂r , e 1 , e 2 , e 3 a local orthonormal basis satisfying ∇ r e i e j = 0 (2.10) at a fixed point (r, x) and also ∇ ∂ ∂r e ≡ 0 (2.11) Finally, given ω 1 , ω 2 1-forms let Using this frame the symmetric two-tensors naturally decompose into three distinct types. In the next four lemmas we record the action of divergence and the Laplacian on these tensors. We will need these preliminary calculations to calculate the action of the Bilaplacian on each of these types of tensors. The proofs of these lemmas can all be found in [1]. (2.5) and (2.6). Then (2.5) and (2.6). Then  (2.5) and (2.6). Then Proof. This will be a straightforward calculation using the three above lemmas. First of all using (2.13) we see that Now it is clear that ∇ * ∇B satisfies (2.5) and (2.6). Thus apply (2.13) again to compute Note that we have applied lemma 6.2 to commute δ past ∇ * ∇. Now, since η satisfies (2.5) and (2.6), so does δB, thus we may apply (2.14) to conclude Similarly we apply (2.16) to conclude Proposition: 2.6. Consider k(r)τ ⊠ dr where τ satisfies (2.5) and (2.6). Then Proof. We must start from the expression in (2.14) and apply (2.13), where we have used the equations from lemma 6.2. Now we apply (2.14) to conclude And similarly And again where we commuted derivatives using lemma 6.2. Finally using (2.16) with l = r −2 k and φ = δτ we compute Collecting together (2.27) -(2.31) gives the result.
Proof. We need to compute the action of the Laplacian of the right hand side of (2.16). First of all it is clear that φ g satisfies (2.5) and (2.6) thus we may apply (2.13) to conclude where we used that δ g = 0 and tr g = 3. Next we note that dφ satisfies (2.5) and (2.6) so that we can apply (2.14) to conclude Where we have used the equation And similarly Finally we apply (2.16) to conclude Collecting together (2.33) -(2.37) gives the result.

The Linearized Equation
In this section we derive the equation for the linearization of the Bach-flat condition at a flat metric. The equation is of course not strictly elliptic due to the diffeomorphism invariance of the Bach-flat equation. After restricting to the case where the variation of the metric is trace-free and divergence-free, we are reduced to the Biharmonic equation. We can then use the decomposition of the Bilaplacian computed in the previous section to classify solutions to this equation and moreover compute the exact decay rates of solutions, which is the key ingredient of our removal of singularities result.
Proposition: 3.1. Given g a flat metric, the linearization of the Bachflat equation at g is equivalent to the equation Proof. Let g(s) be a family of metrics such that g(0) = g and ∂ ∂s g(s) |s=0 = h. First of all since the metric is flat it is clear that As a consequence of the Bianchi identity, in dimension 4 we have the equation where A ij are the components of the Weyl-Schouten tensor A further application of the Bianchi identity then yields The result follows. Now, by a diffeomorphism gauge-fixing procedure which we will use in the applications, we may assume that h comes to us almost divergence-free. In fact we cannot prescribe that h is exactly divergence free. We will address this difficulty later in this section. Also, because of the conformal invariance of the Bach-flat equation, the trace of the right hand side of (3.1) vanishes. Thus there is no a-priori separate equation that the trace of h must satisfy. However, using this conformal invariance in the applications, we can assume that our Bach-flat metric has constant scalar curvature, which in the asymptotically flat regime corresponds to the equation As it turns out we can now remove the trace entirely by adding a term L X g for some vector field X. This corresponds to solving for a vector field X satisfying to preserve the divergence-free condition and where u is a prescribed harmonic function. It is shown in [1] lines 4.11-4.20 that on a Ricci-flat cone one can indeed solve for this X. Thus a divergence-free solution to (3.1) satisfying (3.2) can be written as We now proceed to analyse solutions to (3.5).
Now we would like to write down and simplify the vertical component of equation (3.5). Using propositions 2.20, 2.26 and 2.32 we see that this equation gives Now we want to use equations (3.6), (3.7) and (3.8) to simplify this further. First of all using (3.6) it is clear that Finally using both (3.7) and (3.8) we compute Thus plugging (3.10) -(3.12) into (3.9) gives the equation We would now like to do the same for the cross component of (3.5). First of all from propositions 2.20, 2.26 and 2.32 we see that this is the equation Now, using (3.7) we see that Finally we need to simplify the horizontal component of (3.5). Again using propositions 2.20, 2.26 and 2.32 we get the equation gives Lemma: 3.2. The system of equations (3.5) is equivalent to We now analyse all solutions to this system of equations. We notice that for any solution, l satisfies the determined ODE in (3.26). Thus we can compute the solutions to (3.26) in terms of the eigenvalues of the Laplacian on S 3 acting on functions. Once these are determined we notice that the equations (3.27) and (3.28) show that both f and k will have the same decay rate as l. Once this is done we restrict to the case where l = 0. Here we notice that k satisfies the now determined ODE in (3.25). We can classify these solutions in terms of the eigenvalues of the Laplacian on S 3 acting on one-forms. Once again we can use equation (3.27) to conclude that f has the same decay rate as k. Finally we restrict to the case where l = k = 0. Then it is clear that f satisfies the determined ODE in (3.24), and classify solutions in terms of the eigenvalues of the Laplacian on S 3 acting on traceless symmetric twotensors. We now make this rigorous in a series of lemmas.
Lemma: 3.3. The solutions to (3.5) Proof. Since k = l = 0 we have trB = δB = 0. Say λ is an eigenvalue for the Laplacian of S 3 acting on traceless symmetric two-tensors, and suppose B λ is in the eigenspace of λ. Equation Proof. Note that since l = 0 equation (3.28) implies in that δτ = 0. The eigenvalues for the Laplacian acting on 1-forms on S 3 are given by [5] a j := (j + 1)(j + 3) (3.30) Say τ j is in the eigenspace of a j . Then (3.25) reduces to an ODE with solutions We point out that the decay rates are all less than −2. Also, since the decay rate −4 does not occur, the expression k ′ + 4r −1 k never vanishes, thus by equation (3.27) we see that f = cr −2± √ a j +2 , and the result follows.
Lemma: 3.5. The solutions to (3.5) Proof. The eigenvalues of the Laplacian acting on functions on S 3 are a j := j(j + 2) (3.32) Suppose φ j is in the eigenspace of a j . Equation (3.26) reduces to an ODE with solutions In the case where l(r) = r −3± √ a j +1 , the solution to the whole system is given by a Lie derivative term, specifically L X g 0 where For the solutions l(r) = r −1± √ a j +1 we note that for j > 0 the decay rates are always less than or equal to −2. In the case j = 0, i.e. where φ = c, consider the radially parallel solution, i.e. where l(r) = c. Using equation (3.28) we see that k is also a nonzero constant. Using this equation (3.26) reduces to ∇ * ∇ 2 − 4 τ = 0 so we see that this solution does not in fact occur since 2 is not an eigenvalue of the Laplacian on S 3 acting on one-forms. In any of the cases above, using equation (3.6) we see that f will have the same decay rate as l. Given this, equation (3.27) implies that k must also have the same decay rate.
Together the above lemmas and earlier calculations prove the following proposition.
Proposition: 3.6. On a flat cone, solutions of (3.1) satisfying δh = 0 and ∆ tr h = 0 can be written uniquely as a sum We hasten to point out that we will not in fact be able to guarantee the divergence-free condition. This is due to the presence of certain eigenvalues of the Laplace-Beltrami operator acting on oneforms. Thus we follow the technique used in [1] and consider a modified divergence-free condition. In particular, fix t = 0 with |t| very small. Let We will be able to prescribe that δ t h = 0 for t arbitrarily small. Given this, we define the following modified equation Following the analysis of proposition 3.6 we can write the solutions satisfying ∆ tr h = 0 as growth and decay solutions where the rates are perturbations of those calculated above. In particular we can write where T i is some symmetric bilinear form and {T i } are orthonormal with respect to the inner product We also have the following corollary.
Corollary: 3.7. For t sufficiently small, there are no radially parallel solutions of (3.37) satisfying ∆ tr h = 0.
Proof. We note that the radially parallel solutions found above all had a dr component, and so in the perturbed equation are no longer radially parallel. Thus the only possibility would be f (r)B where f (r) is a constant function and B is trace and divergence-free. However, no such solution occurs according to our analysis above. Thus the corollary follows.
So, according to the above results we can decompose any solution satisfying ∆ tr h = 0 as where h ↑ are the solutions with positive growth rate and likewise h ↓ are solutions which decay in r. Now let We can now state the first of our decay estimates. We start with some notation. For a fixed annulus A a,b (p) we have the norm where the norm || || is defined in 3.39 This norm is defined so that if w = a −2 ψ * a (h), where ψ a is the natural scaling map ψ a (x, r) = (x, ar), then |||h||| a,La = |||w||| 1,L (3.43) Corollary: 3.8. Given 0 < β ′ < β there exists l such that for all a > 0 and L ≥ l

Asymptotic Curvature Decay
In this section we give the proof of theorem 1.1. The proof will follow the techniques used in [1]. We first recall the definition of certain norms introduced in [1]. Let A u denote the natural action of the scaling ψ u on tensors of type (p, q), i.e.
Given T a tensor of type (p, q) we have where the norm on the right is the C k,α -norm with respect to g 0 at the point (1, x). Using this we define And more generally Given T a tensor of type (p, q) we write T ∈ T p,q k,α;l if |T | k,α;l < ∞. We will make use of a gauge-fixing theorem ([1] Theorem 3.1). For a given cone with cone-point p, let A c,d = {(r, x)|c < r < d}.
and δ t (φ * g − g 0 ) = 0 (4.6) Since we have assumed our given Bach-flat metric is asymptotically flat, we can use this theorem to construct gauges relative to the flat metric on the cone in the asymptotic regime. Specifically, let g 0 be the flat metric on R 4 and let g be a given Bach-flat metric on this cone. Consider afixed annulus A c,d and let where φ is from proposition 4.1 with respect to this fixed annulus. Then we note that h satisfies δ t h = 0 (4.8) and further which is a nonlinear elliptic equation on h which we can think of as a perturbation of the linearized deformation equation for h small. The following lemma makes this precise.
Proof. Suppose one had a sequence of gauges φ i and solutions h i where |h i | k,α;0 → 0 but none of the assertions of the proposition hold. By rescaling our solutions and using a compactness argument (see [1] lemma 5.22) we can produce a solution to (3.37) satisfying ∆ tr h = 0 which contradicts lemma 3.9.
Proof of Theorem 1.1 We will adopt the notation used in this section. In particular choose χ and L as in proposition 4.5. Let g 0 denote the standard flat metric on R 4 . Since we already know that the metric g is ALE of order 0, there exists Ψ : R 4 → X so that Now consider the sequence of annuli A L i a,L i+1 a (0). Using (4.21) we may apply proposition 4.1 and choose a δ t -free gauge for Ψ * g with respect to g 0 , φ i , on each annulus. By pushing forward this is equivalent to finding a sequence of flat metrics g i where Ψ * g−g i is δ t -free with respect to g i .
Indeed, if this were not the case then inductively applying proposition 4.5, and in particular using that (4.17) implies (4.5) we can contradict (4.21). Since for any of the flat metrics g i , there are no radially parallel solutions to the linearized deformation equation, we can conclude that (4.20) holds for all i.
Also, by passing to a subsequence, we can assume that for some flat metric g ∞ we have lim j→∞ |g i − g ∞ | k,α ′ ;0 = 0 (α ′ < α) (4.22) Using this together with (4.20) and lemma 4.4 we conclude that This proves that g is ALE of order β ′ .
To prove the full claim we will use an inductive procedure choosing better and better gauges. First of all we point out that the decay estimate (4.23) in fact suffices to find a completely divergence-free global gauge φ (Remark 3.23 [1]). So, let h = φ * g − g 0 be divergence-free. First we get a better estimate on the trace of h. In particular, by lemma 4.3 we have ∆ tr h = F ′ (h) (4.24) where the estimate (4.23) implies Applying the Greens function we can conclude the existence of a function f so that  Again using a Greens function, this decay is sufficient to conclude the existence of h 1 such that Thus P(h−h 1 ) = 0. Given our estimate on the trace of h we can assume that h − h 1 is trace-free. We note that P acting on the traceless piece of h − h 1 also vanishes. Using proposition 3.6 we can write where h 2 ∈ T 0,2 k,α;− min{4β,2} (4.36) and also X ∈ T 1,0 k+1,α;1−β (4.37) Let K X be the diffeomorphism generated by taking the flow of X to time 1. It is clear from the above estimates that (4.38) Furthermore putting together the above estimates we get If 4β ≥ 2 we are done. If not, start over with h = K * −X φ * g − g 0 and proceed as above with β replaced by 4β − ǫ for very small ǫ > 0. It is clear that by induction the result follows.
Remark: The result holds assuming harmonic curvature instead of the Bach-flat condition. In paticular, using the Bianchi identity equation (1.12) implies an equation of the form ∆ Rc = Rm * Rm. The linearized deformation equation is the same as the one analyzed in section 3, and so one can apply the argument in theorem 1.1 to conclude that the metric is in fact ALE of order τ for any τ < 2.

Proof of Theorem 1.2
Let (X, d, x) and B(x, 1)\{x} be as in the statement of the theorem. We already know by theorem 1.1 of [7] that B(x, 1) has a C 0 -orbifold structure at x. Thus in particular, lifting to the universal cover we may suppose x = 0 and B(x, 1) ⊂ D(0, 1) where D denotes the distance ball in the Euclidean metric. Let g be this metric which is smooth away from the origin and C 0 on B(0, 1). Fix a constant s < 1, and consider a sequence of annuli in the metric g be the inversion through the unit sphere and let A ′ i = Φ * A i , and g ′ = Φ * g. Let ρ i denote the distance function of the metric g ′ on the annulus A ′ i . Since we have assumed a bound on the Sobolev constant, the L 2norm of curvature, and that the first Betti number is bounded, we may apply the analysis of section 4 in [7]. In particular we may conclude that ρ i → r as i → ∞. In particular the annuli A ′ i are approaching the standard annuli in R 4 . Thus the asymptotic analysis of section 4 above applies so we conclude that this metric is ALE of order τ for any τ < 2. Pulling this estimate back to B using the spherical inversion gives For δ suficiently small this implies that the metric has a C 1,α extension through the origin. Using results from [3] this implies the existence of a harmonic coordinate around the origin. A simple computation using the Bianchi identity shows that the Bach-flat condition implies that the curvature satisfies an equation of the form ∆ Rc = Rm * Rc. Thus we view the Bach-flat equation in harmonic coordinates as the system ∆ Rc = Rm * Rc (5.3) ∆g = Rc +Q(g, ∂g) (5.4) Using the curvature estimate |Rm| = O(r −δ ) it follows from equation (5.3) that Rc ∈ W 2,p for any p. Using equation (5.4) it is clear that g ∈ W 3,p . This allows us to bootstrap and conclude that g ∈ C ∞ .
Remark: Again we point out that the result holds assuming harmonic curvature instead of the Bach-flat condition. The crucial step in the above argument is the curvature decay rate, and we mentioned after the proof of theorem 1.1 that metrics with harmonic curvature satisfy this estimate. 6. Appendix: Analysis on the Three Sphere Lemma: 6.1. Let (S 3 , g) be the round three-sphere. The smallest eigenvalue of the rough Laplacian acting on traceless divergence-free symmetric two-tensors is 6.
Proof. Using the identification of S 3 with SU(2) we consider the standard global left-invariant Milnor frame X 1 , X 2 , X 3 with structure constants C 1 23 = C 2 31 = C 3 12 = −2 If e 1 , e 2 , e 3 denotes the corresponding coframe, we have the equation where σ(ijk) denotes the sign of the permutation (ijk) and is zero if any of i, j and k are equal. Using this global frame we may write any traceless symmetric two-tensor as B ij e i ⊠ e j and compute ∇ * ∇B = − B kl,ii e k ⊠ e l − 2B kl,i (∇ X i e k ⊠ e l + e k ⊠ ∇ X i e l ) − B kl (∇ X i ∇ X i e k ⊠ e l + 2∇ X i e k ⊠ ∇ X i e l + e k ⊠ ∇ X i ∇ X i e l ) Now a basic calculation using (6.1) shows that ∇ X i ∇ X i e k = −2e k and similarly, using that B is traceless it is easy to compute 2B kl ∇ X i e k ⊠ ∇ X i e l = −2B kl e k ⊠ e l Using these calculations and the fact that the smallest eigenvalue must occur when the coefficient functions are constant, the result follows.