Foliation of an Asymptotically Flat End by Critical Capacitors

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.


Introduction
Riemannian manifolds with asymptotically flat ends play an important role in general relativity and cosmology, and so their general properties are of great interest. In particular, it is often useful to foliate an asymptotically flat end with special surfaces. Huisken and Yau [5] famously proved one can foliate a three-dimensional, asymptotically flat end with constant mean curvature spheres. Furthermore they prove these spheres share a common center, which one can take as the physical center of mass of the system. Previously, R. Ye [14] had shown one can foliate an asymptotically flat end in any dimension n ≥ 3 provided the mass at infinity is nonzero. Subsequently, others have found special foliations by constant expansion surfaces [8], by Willmore surfaces [7], and by isoperimetric surfaces [2]. Here we investigate surfaces which are critical points of the Newton capacity. Recall that, if K ⊂ R n , with n ≥ 3, is a compact set with Lipschitz boundary, one can define its Newton capacity as where ω n is the Euclidean volume of an n-dimensional unit ball and H 1 (R n ) is the Sobolev space of functions with one weak derivative in L 2 (R n ). Standard results in potential theory imply this infimum is realized by the equilibrium potential function U K , which solves the boundary value problem where 0 is the usual, flat Laplacian. Moreover, the solution to (1.2) is unique among all functions which satisfy an appropriate decay condition. It is straight-forward to generalize both (1.1) and (1.2) to the setting of a compact set K in a complete, noncompact Riemannian manifold (M, g) with an asymptotically flat end. The functional Cap is not scale-invariant in Euclidean space, one should not expect it to have critical points as a domain functional. Thus it is natural to seek critical, and even extremal, domains either subject to a constraint or of a modified functional which is scale invariant. One can normalize Cap using the volume of K or the surface area of ∂ K ; both choices are natural and have roots in physics and potential theory [4]. Below we will seek critical sets of a volume-normalized functional, which leads us to the over-determined boundary value problem 0 u = 0 on R n \K , where is a constant. See Sect. A for a derivation of (1.3) as the Euler-Lagrange equation of our normalized domain functional. This computation is standard, but we include it in Appendix A for the reader's convenience. Of course, one can also choose to normalize using the surface area of ∂ K , which leads one to a slightly different over-determined boundary value problem, namely 0 u = 0 on R n \K , where H is the mean curvature of ∂ K . We derive this Euler-Lagrange equation as well, even though we do not require it. A classical theorem of Serrin [12] implies that the only solutions of the overdetermined boundary value problem (1.3) in Euclidean space are round spheres, but one expects the situation to be more complicated in a general Riemannian manifold.
Our setting is that of a Riemannian manifold (M, g) of dimension n ≥ 3 with one asymptotically flat end. In general, the Riemannian manifold (M, g) has one end that is asymptotically flat to order τ > 0 if there exists a compact set K ⊂ M and a diffeomorphism : R n \B → M\K , (1.4) such that in these coordinates where B is the unit ball in R n and ∂ k refers to any collection of partial derivatives of order less than on equal to k. For our main result, we will require the metric to decay more rapidly to the flat metric, so that holds for some σ ∈ R. Our decay condition (1.5) will in particular imply the ADM mass of the flat end (which is defined in the next paragraph below) is zero. Despite its origins in classical potential theory, capacity appears naturally in the context of the geometry associated to scalar curvature and asymptotically flat manifolds. Schoen and Yau first discussed this connection in [11]. Thereafter Bray [1] proved the inequality: if (M, g) is a 3-dimensional Riemannian manifold with R g ≥ 0 and one asymptotically flat end of order τ > 1/2 then Most recently, J. Jauregui [6] defined a capacitory version of the mass of a threedimensional asymptotically flat end as where the outer supremum is over all possible compact exhaustions of M. Jauregui [6] further showed that if (M, g) is a three-dimensional manifold with one asymptotically flat end and R g ≥ 0 then m CV ≥ m ADM . Moreover, if (M, g) is harmonically flat then m CV = m ADM . In the same paper Jauregui conjectures that such an asymptotically flat end carries a foliation by critical points of the capacity function. Our main theorem provides some evidence towards this conjecture, at least for metrics with stronger asymptotics.
Theorem 1 Let (M, g) be a Riemannian manifold of dimension n ≥ 3 with one asymptotically flat end M\K , parameterized as in (1.4) and (1.5). Then there exists ρ 0 > 1 and compact sets K ρ indexed by ρ ∈ (ρ 0 , ∞) such that the domains ρ = M\K ρ are critical capacitors. In other words, there exist functionsū ρ which solve the overdetermined boundary value problem where η is the unit interior normal to K ρ and Our result builds naturally on earlier work, particularly that of the first and second authors [3]. More precisely, they perturb small geodesic balls to produce a family of domains ρ , parameterized by ρ ∈ (0, ρ 0 ), which admit solutions to the overdetermined boundary value problem In our case, the sets K ρ will be perturbations of large coordinate spheres, as defined by the parameterization in (1.4). We end this introduction with a brief outline of the rest of the paper. We begin by reformulating our problem in Sect. 2 to take place on a fixed set. We parameterize this reformulated problem by a radius ρ, a translation τ , and a function w ∈ C 2,α (S). Sect. 3 has some preliminary computations, such as expansions of the metric and the Laplace-Beltrami operators for our reformulated problem, as well as a study of the mapping properties of the Laplace-Beltrami operator on certain weighted function spaces in Sect. 3.3. In Sect. 4 we construct an approximate solution v, given in (4.1), and perturb it by a translation to the eventual solution u ρ,τ,w , given in (4.10). The function u ρ,τ,w already satisfies most of our desired properties: it is harmonic, decays appropriately, and has constant Dirichlet data. It only remains for us to choose parameters ρ, τ , and w so that u ρ,τ,w also has constant Neumann data. To correctly choose these parameters we must invert the Dirichlet-to-Neumann operator of the Laplace-Beltrami operator. We do this in two steps, first writing out an expansion of the normal derivative of u ρ,τ,w and performing a linear analysis of this expansion in Sect. 5, and then completing our nonlinear analysis using the implicit function theorem in Sect. 6. Finally, in Sect. 7 we show that we do in fact produce a foliation of the asymptotically flat end.

Reformulation of the Problem
In this section we reformulate our problem so that we can solve a family of PDEs on the fixed Euclidean domain R n \B. Intuitively, we accomplish three things with this reformulation. First, we rescale by ρ > 0, which one should take to be large. Second, we translate the center of the ball by a small parameter τ ∈ R n . Third, we deform the unit sphere S = ∂B by a function w ∈ C 2,α (S).
We should imagine ρ to be large and both w and τ to be small. So that our parameters are all of the same scale, we require for the remainder of the paper. Putting all these transformations together we obtain a parameterization and let ρ,τ,w = ρ,τ,w (R\B). Finally we can use ρ,w,τ to pull problem (1.6) back to R n \B. Under this change of coordinates, we have now reformulated our original problem (1.6) as where g = * ρ,w,τ (g), η is the inward pointing unit normal to S = ∂B with respect to the metric g. Both our new metric g and the function u ρ,w,τ depend on the three parameters ρ ∈ (0, ∞), w ∈ C 2,α (S), and τ ∈ R n .

Preliminary Computations
In this section we carry out some preliminary computations, in preparation for solving (2.3) We first write out a Taylor expansion of the metric g.

Notation
All our computations in this section are perturbation expansions, when ρ is large and |τ | and w C k,α (S) are small. In the computations below we will sometimes wish to extend a function v defined on the sphere to a tubular neighborhood, and we do so by making it constant in the radial direction, taking w(x) = w x |x| . Similarly, for each w ∈ C 2,α (S) we let To make the computations below tractable, we adopt the following notation throughout the rest of the paper.
For i ∈ {0, 1, 2} we let L i denote a linear partial differential operator of order i whose coefficients depend smoothly on ρ and x and that satisfies the bound Similarly we let Q i denote a nonlinear operator of order i ∈ {0, 1, 2} such that Q i (0, 0) = 0 and that satisfies the bound Finally we let P i be a function of the form for some positive constant depending only on h, n, k, , m, i, α and ρ 0 . For brevity we write It is important to observe that the product of any two terms, each of which has the form of either L i or Q i , has the form of P i .

Metric Expansions and the Laplacian
We have the following expansions.

Lemma 3.1 We have
and where Proof Letting {e 1 , . . . , e n } be the standard orthonormal basis for R n , we see where we evaluate derivatives of at ρτ + ρx + ρw(x/|x|)x. Hence Next, we write |x| = r and so that Observe that we absorb the term (1 − n)ρ 1−n r 1−n w above into L 0 (w), while the corresponding linear term with respect to τ is kept. Indeed when solving the nonlinear equation for small ρ in Sect. 6, we have to replace w with ρ n−1 w , which increases the power of ρ in (1 − n)ρ n−1 r 1−n w by n − 1.
In the next sections, we will work with the metric where 0 is the usual flat Laplacian. Moreover, Proof Recall that the Laplace-Beltrami operator has the form Then using (3.6) and (3.7), we have In addition we have from (3.8) which yield the expansions in Lemma 3.2.

Weighted Spaces
The best setting in which to perform our linear analysis is that of weighted Hölder spaces. Following Pacard and Rivière [9], we use the following definition. Definition 1 Let ν ∈ R, k ∈ N and 0 < α < 1. Then we say u ∈ C k,α Here where A s = {x ∈ R n : s < |x| < 2s}. We denote the space of functions vanishing on the boundary by Intuitively, one can think of C 0,α ν (R n \B) as those functions which grow at most like |x| ν when |x| is large.

Remark 1
Pacard and Rivière perform their analysis on weighted Hölder spaces on B\{0}, whereas we want to examine functions on R n \B. It is straight-forward to transfer between the two settings using the Kelvin transform K, defined by 14) It will be convenient to also note the transformation law One can show the following theorem (see Sect. 2.2 of [9]).

Theorem 2 The mapping
The mapping properties of 0 change when the weight ν crosses over one of the indicial roots γ ± j , where and λ j is the jth eigenvalue of the Laplace-Beltrami operator on the sphere. Thus one can recover the indicial roots γ ± j as growth/decay rates of solutions to the ODE A slightly more refined analysis uncovers the following theorem.
Theorem 3 Let ν < 0 with ν / ∈ {γ ± j : j ∈ N}, and let j 0 be the least non-negative integer such that ν > γ − j 0 . Then the cokernel of the mapping has dimension j 0 . Alternatively let ν > 2 − n with ν / ∈ {γ ± j : j ∈ N}, and let j 0 be the least positive integer such that ν < γ + j 0 then kernel of the mapping Again, we refer the reader to Sect. 2.2 of [9] for details. Replacing ρ by 1/ρ in Lemmas 3.1-3.2 and keeping the notation g ρ for the metric g 1/ρ , we have the following result. Lemma 3.3 Let 2 − n < ν < 0. There exist ρ 0 > 0 and c 0 > 0 such that the map is well defined and smooth. Here B r (0) will refer to a ball centered at the origin of radius r in the appropriate space, either R n or a space of functions depending on the context. Furthermore, for every (ρ, w, τ ) is invertible and for all u ∈ C 2,α ν,D (R n \B) we have the inequalities

Approximate and Actual Solutions
In this section we construct an approximate solution using the standard Greens function in Euclidean space and compare it to the solution of a corresponding Dirichlet problem.
We have the following expansion.

Lemma 4.1 For ρ sufficiently small, the Laplacian of v is given by
and so With this, we have The expansion in Lemma 4.1 now follows from Lemma 3.2 after replacing ρ by 1/ρ.

Expansion of the Normal Derivative and Linear Analysis
The aim of this section is to derive an expansion of the normal derivative of the solution u ρ,τ,w . We start by the computation of the interior unit normal vector to B . The interior unit normal vector field to B with respect to the metric g ρ is given by a k k and a k = ∇ S w, k + P 1 (ρ, , w, τ ).
Proof For each ∈ S, the vectors span the tangent space T S and since T S R n−1 , we may assume after orthonormalization that i , j = δ i j . In addition having , = 1, it follows that , = 0 and ∇ S w, g S = 0 on S. With these indentities, Lemma 3.1 yields and , j g ρ = P 1 (ρ, , w, τ ).
We look for a normal vector η of S with respect to the metric g in the form The condition that η is normal is thus equivalent to η, (5.6) By Lemma 3.1, We also have Now using (5.6) and (5.8), Next we compute We have by Lemma 3.1, and from (5.4), In addition, where we have used (5.11) to get the last equality. Using (5.10) and (5.9), it follows that We gather (5.13), (5.14) and (5.15) to obtain η, η g ρ = 1 + 2w + σ |z| 1−n + P 1 (ρ, , w, τ ) .
The normal interior unit vector field to B for the metric g ρ is then given by (5.16) and (5.1) follows from (3.8).

Expansion of the Normal Derivative
The following proposition yields the expansion of the normal derivative of u ρ,τ,w with respect to the metric g ρ .

Proposition 4
For ρ sufficiently small the normal derivative of u = u ρ,τ,w = v + ρ,τ w with respect to the metric g ρ on ∂B is given by H (ρ, τ, w), and ρ,τ,w is solution of (4.7).

Linear Analysis of the Normal Derivative
In this section, we analyse (5.17) and (5.18) in detail and compute the operators that will allow us to solve and apply the implicit function theorem to find τ and w as functions of a small variable ρ. As shown in Proposition 5 below, the linearised operator of H with respect to w has a non trivial kernel. To solve (5.25), we then need to project on this kernel and first express τ in terms of ρ and w. Still there are some challenges to overcome. Indeed, since τ appears with a factor of ρ n−1 in (5.18) and (4.2), the derivative of H with respect to τ evaluated at (0, 0, 0) is zero. We compensate for this by multiplying the equation ( where ρ,τ,w := ρ 1−n ρ,τ,ρ n−1 w . We emphasise that the function ρ,τ,ρ n−1 w is the unique solution of (4.7) with w replaced by ρ n−1 w in both the metric coefficients of g ρ involved in the operator g ρ , and the right hand side of the interior and boundary equations. We keep the notation g ρ for this new Laplacian. The quantity ν i g ρ (ρ n−1 w) is the ith-component of the unit vector ν g ρ in (5.17) where w is also replaced with ρ n−1 w.
By Lemma 4.1 that ρ,τ,w is solution the unique solution of In addition, 0,0,0 = C 1 (n, σ )K, (5.29) where C 1 (n, σ ) = − (n − 1)(n − 2) 2 2 σ and K is the unique solution of It is plain that the function K is radial, and one obtain by Gauss-Green formula ∂ ν K = −K (1) = 1 n−1 . Now from (5.29) and (5.27) it follows that The non linear problem under consideration is then to show that provided ρ is small, we can find τ and w such that We start by studying the operator G. and consider the smooth map T : R * + × R * + × U → C 0,α (∂B) defined by For every ω ∈ C 2,α (∂B) we have As already explained before (5.28), w is replaced by ρ n−1 w in the metric coefficients of g ρ involved in g ρ . Hence evaluating (5.37) at (ρ, τ, w) = (0, 0, 0) and using (3.12) and Lemmas 3.1-3.2 we find after differentiating the right hand of (5.28) that the function which is the desired expression in (5.35).
Next we prove (5.34). We use a similar argument as above and differentiate (5.28) with respect to τ at (ρ, τ, w) = (0, 0, 0). This yields a function Next, we show that be the orthogonal projection on span{x 1 ; . . . ; x n } and consider the function We know that 0 X i = 0 in R n \ {0}. We multiply (5.41) with X i and integrate by parts to get This implies that From this, we deduce that We claim that To see this, we let Y k i be the spherical harmonics for which Y 1 i = x i for i = 1, . . . , n, corresponding to the eigenvalues k(k + n − 2) on sphere. We suppose that k = 1. We then define Then X k i are admissible test functions in (5.41). We observe that the right hand side in (5.41) is in L 2 (R n \B). Therefore by simple arguments, we have that F τ ∈ H 1 (R n \B). Using the decomposition in spherical harmonics of F τ , we can see that F τ (x) = f (|x|) x, τ , for some some function f . From this we can multiply (5.41) by X k i and use the Gauss-Green formula to deduce that as claimed. Gathering (5.42), (5.43) and (5.44), we obtain (5.34).
Therefore lim |x|→0 |x| n−2 K(ψ w )(x) = 0, so the origin is a removable singularity for K(ψ w ) and thus K(ψ w ) solves (5.49) By elliptic regularity theory, K(ψ w ) ∈ C 2,α (B). From (5.48) and the definition of L we get Thanks to [10], the spectrum of the operator L is given by meaning that the kernel of the operator L is given by the space V 1 spanned by linear coordinates on the sphere S Moreover there exists a constant C > 0 such that provided w ∈ ⊥ C ,α (S).

Solving the Nonlinear Problem
We want to show that provided ρ is small, we can find τ and w such that (5.32) holds. That is Then from (5.31), We denote by the L 2 -orthogonal projection from C 1,α (S) onto V 1 and T : V 1 → R n the isomorphism sending x i ∂B to e i . We also define := T • , and consider the equation The mapping K has the following properties: • K (0, 0, 0) = 0. This is from (6.2), • D τ (ρ,τ,w)=(0,0,0) K is a the identity in R n times a constant, which follows from (5.34).
for any R > 0, and a quick compation gives Cap(B R ) = R n−2 , with the equilibrium potention function U (x) = R n−2 |x| 2−n . As we discussed in the introduction, one can either normalize using volume or surface area, leading to the following two scaleinvariant functionals: