On far-outlying CMC spheres in asymptotically flat Riemannian $3$-manifolds

We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the"far-off-center"regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on the behavior of scalar curvature at infinity.


Introduction
We complement our recent work [5] on the characterization of the leaves of the canonical foliation as the unique large closed embedded stable constant mean curvature surfaces in strongly asymptotically flat Riemannian 3-manifolds. More precisely, we extend here the Lyapunov-Schmidt analysis of outlying stable constant mean curvature spheres that developed by S. Brendle and the secondnamed author in [3] to also include the far-off-center regime and general Schwarzschild asymptotics.
We begin by introducing some standard notation.
Throughout this paper, we consider complete Riemannian 3-manifolds (M, g) so there are both a compact set K ⊂ M and a diffeomorphism M \ K ∼ = {x ∈ R 3 : |x| > 1/2} such that in this chart at infinity, for some q > 1/2 and non-negative integer k, where for all multi-indices I of length |I| ≤ k. Moreover, we require that the boundary ∂M of M , if non-empty, is a minimal surface, and that the components of ∂M are the only connected closed minimal surfaces in (M, g). We say that (M, g) is C k -asymptotically flat of rate q.
It is convenient to denote, for r > 1, by S r the surface in M corresponding to the centered coordinate sphere S r (0) = {x ∈ R 3 : |x| = r}, and by B r the bounded open region enclosed by S r and ∂M . Given A ⊂ M , we let r 0 (A) := sup{r > 1 : B r ⊂ A}. 1 A particularly important example of an asymptotically flat Riemannian 3-manifold is Schwarzschild initial data M = {x ∈ R 3 : |x| ≥ m/2} and g = 1 + m 2|x| where m > 0 is the mass parameter. We say that (M, g) as above is C k -asymptotic to Schwarzschild of mass m > 0, if, instead of (1), we have where for all multi-indices I of length |I| ≤ k. The contributions in this paper combined with the key result in [5] lead to the following theorem.
Theorem 1.1 ( [5]). Let (M, g) be a complete Riemannian 3-manifold that is C 6 -asymptotic to Schwarzschild of mass m > 0 and whose scalar curvature vanishes. Every connected closed embedded stable constant mean curvature surface with sufficiently large area is a leaf of the canonical foliation.
The canonical foliation {Σ H } 0<H<H 0 of M \ K through stable constant mean curvature spheres Σ H (with respective mean curvature H) was discovered by G. Huisken and S.-T. Yau [8]. They show that, for every s ∈ (1/2, 1], there is H s ∈ (0, H 0 ) such that Σ H for H ∈ (0, H s ) is the only stable constant mean curvature sphere of mean curvature H in (M, g) that encloses the ball {x ∈ R 3 : |x| < H −s } in the chart at infinity. This characterization of the leaves was later refined by J. Qing and G. Tian [12]: Upon enlarging K and shrinking H 0 > 0 accordingly, if necessary, each Σ H of the canonical foliation {Σ H } 0<H<H 0 is the unique stable constant mean curvature sphere of mean curvature H in (M, g) that encloses K. In joint work with A. Carlotto [4], inspired by earlier work of J. Metzger and the second-named author [6], we have extended this characterization further under the additional assumption that the scalar curvature of (M, g) is non-negative in the following way: Choose a point p ∈ M . Every connected stable constant mean curvature sphere Σ ⊂ M that encloses p and whose area is sufficiently large is a leaf of the canonical foliation. Thus, to prove an unconditional uniqueness result along the lines of Theorem 1.1, it remains to understand large stable constant mean curvature spheres that are outlying in the sense that the region they enclose is disjoint and -in view of the results in [4] -far from K. The center of mass flux integrals used in [8,12] as a centering device vanish in this case regime; new ideas are needed. S. Brendle and the second-named author [3] have discovered a subtle relationship between scalar curvature and outlying stable constant mean curvature spheres. They give examples of divergent sequences {Σ k } ∞ k=1 of outlying stable constant mean curvature spheres in (M, g) asymptotic to Schwarzschild with m > 0, which is the setting of [8,12]. In fact, Σ k is a perturbation of the coordinate sphere in the chart at infinity, where ξ ∈ R 3 is such that |ξ| > 1 and λ k → ∞. On the other hand, they show that no such sequences can exist in (M, g) if the scalar curvature is non-negative, provided a further technical assumption on the expansion of the metric in the chart at infinity holds. Theorem 1.2 (S. Brendle and M. Eichmair [3]). Let (M, g) be a complete Riemannian 3-manifold that is C 4 -asymptotic to Schwarzschild with mass m > 0, where, in addition to (2), we also ask that with corresponding estimates for all partial derivatives of order ≤ 4, and where T ij is homogeneous of degree −2. There does not exist a sequence of outlying stable constant mean curvature surfaces {Σ k ⊂ M } ∞ k=1 whose inner radius r 0 (Σ k ) and mean curvature H(Σ k ) satisfy In our recent work [5], we show that when (M, g) is asymptotic to Schwarzschild with mass m > 0 and if the scalar curvature is non-negative, there are no sequences of embedded stable constant mean curvature spheres Assuming in addition that the metric has the form in Theorem 1.2, this leaves only the case of To rule out this scenario, we revisit the Lyapunov-Schmidt reduction in [3]. Our other main goal here is to investigate whether top-order homogeneity in the expansion of the metric (3) off of Schwarzschild in Theorem 1.2 is really necessary. Neither the results [8,12,4] for spheres that are not outlying nor the main result of [5] require such an assumption. It turns out that Theorem 1.2 is false without additional such conditions. Theorem 1.3. There is an asymptotically flat complete Riemannian 3-manifold (M, g) with nonnegative scalar curvature that is smoothly asymptotic to Schwarzschild of mass m > 0 in the sense that for all multi-indices I, which contains a sequence of outlying stable constant mean curvature spheres It turns out that it is possible to recover a version of Theorem 1.2 without demanding homogeneity in the expansion of the metric if instead we impose a mild growth condition on the scalar curvature. Theorem 1.4. Let (M, g) be a complete Riemannian 3-manifold that is C 4 -asymptotic to Schwarzschild in the sense that for all multi-indices I of length |I| ≤ 4. We also assume that either There does not exist a sequence of outlying stable constant mean curvature surfaces Σ k ⊂ M whose inner radius r 0 (Σ k ) and mean curvature H(Σ k ) satisfy Note that (5) holds in either one of the following two cases.
(i) When R = 0. This is for example the case when (M, g) is time symmetric initial data for a vacuum spacetime. (ii) When the metric in the chart at infinity has the special form (3) in Theorem 1.2, then It is interesting to compare (5) to condition (H3) in S. Brendle's version of Alexandrov's theorem for certain warped products [1]. We remark that the example constructed in Theorem 1.3 is a warped product. We also mention that S. Ma has constructed examples of (M, g) that contain large unstable constant mean curvature spheres [9]. The scalar curvature in these examples is negative in some places; see the discussion preceding the statement of Theorem 1.1 in [9] and the proof of Lemma 4.7 therein. We now turn to the case of surfaces that are very far outlying in the sense that These surfaces are not within the scope of the Lyapunov-Schmidt reduction carried out in [3], where the case (4) is considered. The main difficulty in this regime is that the "Schwarzschild contribution" to the reduced area functional leveraged in [3] is no longer on the order of O(1), but is instead decaying. As such, it is necessary to obtain rather involved estimates for the reduced functional. To describe our results, we first recall some terminology from [3] that we will also adopt. A standard application of the implicit function theorem gives that, for λ > 0 and ξ ∈ R 3 large, we can find closed surfaces Σ (ξ,λ) in the chart at infinity so that the following hold: • Σ (ξ,λ) bounds volume 4πλ 3 /3 with respect to the metric g.
• u (ξ,λ) is orthogonal to the first spherical harmonics on S λ (λ ξ) with respect to the Euclidean metric. • The mean curvature of Σ (ξ,λ) with respect to g viewed as a function on S λ (λ ξ) is the restriction of a linear function.
Given a sequence of connected closed stable constant mean curvature surfaces {Σ k } ∞ k=1 with r 0 (Σ k ) → ∞ and r 0 (Σ k )H(Σ k ) → ∞, the same argument as in [3, p. 676] shows that Σ k = Σ (ξ k ,λ k ) for appropriate λ k > 0 and ξ k ∈ R 3 when k is sufficiently large. Note that λ k > 0 and ξ k ∈ R 3 are both large in this case. Whether (M, g) admits such sequences can now be decided using the following result.
Theorem 1.5. Let (M, g) be a complete Riemannian 3-manifold that is C 5+ℓ -asymptotic to Schwarzschild with mass m = 2, where ℓ ≥ 0 is an integer. Let λ > 0 and ξ ∈ R 3 be large. We have 1 where R is the scalar curvature of (M, g). This expansion can be differentiated ℓ times with respect to ξ.
As in [3], we use that for ℓ ≥ 1, the map ξ → area g (Σ (ξ,λ) ) has a critical point at ξ if, and only if, Σ (ξ,λ) is a constant mean curvature sphere. If ℓ ≥ 2, then the critical point is stable if, and only if, Σ (ξ,λ) is a stable constant mean curvature sphere. This immediately leads to the following corollary. Corollary 1.6. Let (M, g) be a complete Riemannian 3-manifold that is C 6 -asymptotic to Schwarzschild in the sense that for all multi-indices I of length |I| ≤ 6. Assume that the scalar curvature vanishes. There does not exist a sequence of connected closed stable constant mean curvature Lyapunov-Schmidt reduction has also been used by e.g. R. Ye [13], S. Nardulli [10], and F. Pacard and X. Xu in [11] to study when small geodesic spheres admit perturbations to constant 1 We may compute the Laplacian of scalar curvature either with respect to g or with respect to the Euclidean background metric in the chart at infinity. The difference may be absorbed into the error terms of the expansion. mean curvature. S. Nardulli [10] has studied the expansion for small volumes of the isoperimetric profile of a Riemannian manifold.
The analogue of Theorem 1.4 in this setting is not so clear-cut. We have the following result. Corollary 1.7. Let (M, g) be a complete Riemannian 3-manifold that is C 7 -asymptotic to Schwarzschild in the sense that We also assume that the scalar curvature R of (M, g) is radially convex at infinity in the sense that outside of a compact set. There does not exist a sequence of connected closed stable constant mean It turns out that the hypothesis (7) is surprisingly sharp. Comparing with Theorem 1.2 or Theorem 1.4, one might be lead to conjecture that it can be weakened to as |x| → ∞ where T ij homogeneous of order −2, and that the scalar curvature is non-negative.
The second alternative assumption here implies the first -by Euler's theorem.
The following example dashes any hope of such generalizations.
Theorem 1.8. There is an asymptotically flat complete Riemannian 3-manifold (M, g) with nonnegative scalar curvature such that, in the chart at infinity, along with all derivatives, where T ij is homogeneous of degree −2, and which contains outlying stable constant mean curvature spheres Σ k ⊂ M with Finally, we note that there is by now an impressive body of work on stable constant mean curvature spheres in general asymptotically flat Riemannian 3-manifolds. We refer the reader to Section 2.1 in [5] for an overview and references to results in this direction.
loss of generality, we may assume that the mass m is equal to 2. Thus, for all multi-indices I of length |I| ≤ 4.
Let Ω be a bounded subset with compact closure in R 3 \ B 1 (0). For ξ ∈ Ω and λ > 0 sufficiently large, we may use the implicit function theorem to find surfaces Σ (ξ,λ) as in Proposition 4 of [3]. Moreover, the surface Σ (ξ,λ) is a constant mean curvature sphere (respectively, a stable constant mean curvature sphere) if, and only if, ξ is a critical point (respectively, a stable critical point) for the map The derivation of Proposition 5 in [3] carries over to give The assumption that σ is homogeneous is neither needed nor used at this point of [3]. We recall that is the contribution from σ.
Here and below, unless explicitly noted otherwise, all geometric operations are with respect to the Euclidean background metric in the chart at infinity.

Radial variation.
The computation of the radial derivative of (9) in Section 3 of [3] uses the top-order homogeneity of σ that is part of their assumption repeatedly. Here, we compute this derivative in the general case, employing several integration by parts to derive a geometric expression involving the scalar curvature on the nose.
We define a vector field The first variation formula gives We insert this into the above expression, and continue.
We define a vector field W = div σ − ∇ tr σ.
where R is the scalar curvature of g. In conclusion, we obtain This computation connects the radial derivative of F σ with the scalar curvature R of g. We emphasize again that our derivation parallels the proof of Proposition 7 in [3], though we do not assume the top order homogeneity of σ.

Radial variation in spherical coordinates. Assume first that
For definiteness, we assume that ξ = |ξ| e 3 where |ξ| > 1. In this subsection, we compute the radial variation on the complement of the z-axis. The radial line in direction intersects the sphere B λ (λ ξ) in the ρ-interval whose endpoints are the solutions The intersection is non-empty for angles φ ∈ [0, φ + ] where φ + ∈ (0, π) solves We then have that Arguing as in [3, p. 677] shows that Σ (ξ,λ) cannot be a constant mean curvature sphere. We now observe that the above arguments go through under the weaker assumption (5). Indeed, using that R = O(|x| −4 ) from asymptotic flatness, we obtain upon integrating inwards from infinity Under these assumptions, the preceding computation leads to the estimate We also mention that (5) is implied by the assumption both as |x| → ∞. In particular, it follows from the assumptions in Theorem 1.2.

Proof of Theorem 1.3
Our strategy here parallels the proof of Theorem 1 in [3] in that we construct our metric to have a pulse in its scalar curvature, which in turn forces the reduced area functional ξ → area g (Σ (ξ,λ) ) to have stable critical points. Unlike in [3], our examples are spherically symmetric (which also simplifies the analysis) and, more importantly, they have non-negative scalar curvature.
From this, the asserted decay of the higher derivatives can be verified by induction.
On R 3 \ {0}, we define a conformally flat Riemannian metric g = (1 + 1/r + ϕ(r)) 4ḡ = (1 + 1/r) where r = |x|. Note that g is smoothly asymptotic to Schwarzschild with mass 2. Its scalar curvature is easily computed as In particular, it is non-negative on the complement of a compact set. We now make a particular choice for S. Fix χ ∈ C ∞ (R) that is positive on (3,4) and suppored in [3,4]. Let where A > 0 is a large constant that we will fix later. Recall from (8) that We choose ξ ∈ R 3 with 2 ≤ |ξ| ≤ 9 and λ = 10 j where j ≥ 1 is a large integer. Using (10), we compute the radial derivative as When |ξ| = 2 √ 2, the integral on the last line is negative. We choose A > 0 large so that the sum of the first two terms is negative. When |ξ| = 5, the second term vanishes while the first term is strictly positive. Thus, for j ≥ 1 sufficiently large, the derivative d ds s=1 area g (Σ (sξ,λ) ) is negative when |ξ| = 2 √ 2 and positive when |ξ| = 5. Using that the metric g is rotationally symmetric, we see that the map ξ → area g (Σ (ξ,10 j ) ) has a stable critical point (a local minimum) at some ξ j ∈ R 3 with |ξ j | ∈ (2 √ 2, 5). In other words, Σ (ξ j ,10 j ) is a "far-off-center" stable constant mean sphere for j sufficiently large.
Remark 3.2. S. Brendle has already observed in Theorem 1.5 of [1] that, as a consequence of the work by F. Pacard and X. Xu in [11], every rotationally symmetric Riemannian manifold whose scalar curvature has a strict local extremum contains small stable constant mean curvature spheres.

Proof of Theorem 1.5
Consider for all multi-indices I of length |I| ≤ 7.
Our proof is guided by the Lyapunov-Schmidt reduction and the related expansion for the reduced area functional as developed in [3]. The goal is to extend these ideas to allow for ξ → ∞. For a useful analysis in this regime, it is necessary to develop the expansion of the reduced area functional to a higher order than was necessary in [3], which turns out to be quite delicate. Our computations are also related and in part inspired by those for exact Schwarzschild in Appendix A of [2].
Let ξ ∈ R 3 and λ > 0 large. There is r > 1 with r ∼ λ and a smooth function u (ξ,λ) on the sphere S r (λ ξ) that is perpendicular to constants and linear functions with respect to the Euclidean metric and such that the mean curvature with respect to g of the Euclidean normal graph Σ (ξ,λ) of u (ξ,λ) -as a function on S r (λ ξ) -is a linear combination of constants and linear functions and such that vol g (Σ (ξ,λ) ) = 4πλ 3 /3.

Moreover, sup
Sr(λ ξ) This is a standard consequence of the implicit function theorem and elementary analysis; cf. Proposition 4 in [3].
We will improve estimate (12) below.
It is convenient to abbreviate a = λ ξ.
We will frequently use the computations results listed in Appendix A in this section. Estimating vol g (B r (a)). Recall the following expansion for the determinant of a matrix

4.1.
Thus, we have Repeating the computations in Proposition 17 of [2] (noting the dependence of the error on r), we find We now turn to the second term in the expansion of the volume form. We will write σ for σ evaluated at a (we will use the convention that if σ appears with a derivative, the derivative is taken and then the quantity is evaluated at a). First, note that for y ∈ B r (0) with x = a + y, as well as Finally, we have tr σ = tr σ + ∇ y (tr σ) + 1 2 ∇ 2 y,y (tr σ) + 1 6 ∇ 3 y,y,y (tr σ) + 1 24 ∇ 4 y,y,y,y (tr σ) We will frequently consider such Taylor expansions for expressions involving σ.
Combining the above expansions and using the expressions found in Appendix A, we have

Br(a)
tr σ Br ∇ 2 y,y tr σ Br a, y ∇ y tr σ Continuing on, we have that Now, putting these terms together, we find that vol g (B r (a)) = 4π 3
We then compute
For the higher eigenspaces, we will be content with the estimate

4.5.
Estimates for u. Our goal here is to improve upon the initial estimate (12).
Let t ∈ [0, 1]. Consider the Euclidean graph over S r (a) of the function t u. The initial normal speed with respect to g of this family can be computed as w = u g(y/r, ν g ).
Note that w = (1 + O(|x| −1 ))u up to and including second derivatives. We will give a more precise estimate later. Thus, the second variation of area implies that ∆ Sr(a) g w + (|h g | 2 g + Ric g (ν g , ν g ))w = where, as before, H g is the mean curvature of S r (a) with respect to g. It follows that This allows us to improve the coarse estimate above to ∆ Sr(a) g w + (|h g | 2 g + Ric g (ν g , ν g ))w = H g − H Σ g + O(λ −5 |ξ| −4 ) + O(λ −3 |ξ| −6 ).
At this point, we can improve our earlier estimate for w to up to and including second derivatives. Thus Continuing on, we have that and Putting these estimates together, we find that Hence, This implies that together with two derivatives. Note that in particular along with two derivatives. The above expression also implies that with two derivatives.
We have seen above that We begin with the first term.
Sr(a) u σ(y, y) Finally, the last term satisfies Putting this together, we find that Estimating R and ∆ g R. We now relate the previous expression to the scalar curvature R of (M, g). As with mean curvature, we first consider Then, Thus, we find that Similarly, Thus, we find that It follows that Thus, it remains to estimate ∆ĝ|x| −1 . We have that Thus, Thus, we find that Similarly, ∆R = ∆ (div(div(σ)) − ∆ tr σ)
This completes the proof of Theorem 1.5.

5.
Proof of Corollary 1.7 We assume that (M, g) is C 6 -asymptotically Schwarzschild in the sense that where ∂ I σ ij = O(|x| −2−|I| ) for all multi-indices I of length |I| ≤ 6. We also assume that outside of a compact set. This condition integrates to yield We now consider a sequence of connected closed stable constant mean curvature surfaces Σ k with For k large, we may find λ > 0 and ξ ∈ R 3 large so that Σ k = Σ (ξ,λ) . By Theorem 1.5, Using this and (7), we may integrate in the in the radial direction to find that for t ≥ 0, Integrating this again, we find that Choosing t judiciously we arrange for the term in parenthesis to be o(1). We have proven that Now, considering the first variation of area g (Σ (ξ,λ) ) in directions orthogonal to ξ as above. We obtain that the full derivative satisfies On the other hand, because ∂ r R = o(λ −5 |ξ| −5 ), Taylor's theorem combined with ∂ r R ≤ 0 yields Combining this with (7), we obtain Similarly, combining the facts R ≥ 0, R = o(λ −4 |ξ| −4 ), and DR = o(λ −5 |ξ| −5 ) with Taylor's theorem yields Similarly, we find that Finally, since Returning to the radial first variation, we see that This contradiction completes the proof.
6. Proof of Theorem 1.8 As in the proof of Theorem 1.3, our strategy is parallel to the proof of Theorem 1 in [3], except that here we also exploit that the various terms in the reduced area functional ξ → area g (Σ (ξ,λ) ) have different orders in the regime where ξ → ∞.
Appendix A. Some integral expressions In this appendix, we recall several standard identities that are used in the proof of Theorem 1.5.
A.1. Integrals over B r (0 B ijkl y i y j y k y l = 4π 5 If B ijkl is symmetric in the first two slots and in the second two slots separately, we obtain Assume now that the tensor C ijklmn on R 3 is symmetric in the first four indices and, separately, in the last two indices. Then, i,j,k,l,m,n Sr(0) C ijklmn y i y j y k y l y m y n = i C iiiiii A.3. Some useful integrals. The following computations needed in the proof of Theorem 1.5 are readily verified using the identities from the previous subsection.