On far-outlying CMC spheres in asymptotically flat Riemannian 3 -manifolds
aa r X i v : . [ m a t h . DG ] M a r ON FAR-OUTLYING CMC SPHERES IN ASYMPTOTICALLY FLATRIEMANNIAN -MANIFOLDS OTIS CHODOSH AND MICHAEL EICHMAIR
Abstract.
We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the workof S. Brendle and the second-named author [3] to the “far-off-center” regime and to include generalSchwarzschild asymptotics. We obtain sharp existence and non-existence results for large stableCMC 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 foliationas the unique large closed embedded stable constant mean curvature surfaces in strongly asymptot-ically flat Riemannian 3-manifolds. More precisely, we extend here the Lyapunov-Schmidt analysisof outlying stable constant mean curvature spheres that developed by S. Brendle and the second-named 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 botha compact set K ⊂ M and a diffeomorphism M \ K ∼ = { x ∈ R : | x | > / } such that in this chart at infinity , for some q > / k , g ij = δ ij + τ ij (1)where ∂ I τ ij = O ( | x | − q −| I | )for all multi-indices I of length | I | ≤ k . Moreover, we require that the boundary ∂M of M , ifnon-empty, is a minimal surface, and that the components of ∂M are the only connected closedminimal 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 centeredcoordinate sphere S r (0) = { x ∈ R : | x | = r } , and by B r the bounded open region enclosed by S r and ∂M . Given A ⊂ M , we let r ( A ) := sup { r > B r ⊂ A } . A particularly important example of an asymptotically flat Riemannian 3-manifold is Schwarz-schild initial data M = { x ∈ R : | x | ≥ m/ } and g = (cid:16) m | x | (cid:17) X i =1 dx i ⊗ dx i where m > 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 g ij = (cid:16) m | x | (cid:17) δ ij + σ ij (2)where ∂ I σ ij = O ( | x | − −| I | )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 -manifold that is C -asymptotic toSchwarzschild of mass m > and whose scalar curvature vanishes. Every connected closed embeddedstable constant mean curvature surface with sufficiently large area is a leaf of the canonical foliation. The canonical foliation { Σ H }
0, which is the setting of [8, 12]. In fact, Σ k is a perturbation of the coordinate sphere S λ k ( λ k ξ ) = {| x − λ k ξ | = λ k : x ∈ R } AR-OUTLYING CMC SURFACES 3 in the chart at infinity, where ξ ∈ R is such that | ξ | > λ k → ∞ . On the other hand, theyshow that no such sequences can exist in ( M, g ) if the scalar curvature is non-negative, provided afurther 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 -manifoldthat is C -asymptotic to Schwarzschild with mass m > , where, in addition to (2) , we also askthat g ij = (cid:16) m | x | (cid:17) δ ij + T ij + o ( | x | − ) as | x | → ∞ (3) with corresponding estimates for all partial derivatives of order ≤ , and where T ij is homogeneousof degree − . There does not exist a sequence of outlying stable constant mean curvature surfaces { Σ k ⊂ M } ∞ k =1 whose inner radius r (Σ k ) and mean curvature H (Σ k ) satisfy r (Σ k ) → ∞ and r (Σ k ) H (Σ k ) → η > . (4)In our recent work [5], we show that when ( M, g ) is asymptotic to Schwarzschild with mass m > { Σ k } ∞ k =1 in ( M, g ) with r (Σ k ) → ∞ and r (Σ k ) H (Σ k ) → . Assuming in addition that the metric has the form in Theorem 1.2, this leaves only the case of r (Σ k ) → ∞ , area g (Σ k ) → ∞ , r (Σ k ) H (Σ k ) → ∞ . 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 ofthe 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 turnsout that Theorem 1.2 is false without additional such conditions.
Theorem 1.3.
There is an asymptotically flat complete Riemannian -manifold ( M, g ) with non-negative scalar curvature that is smoothly asymptotic to Schwarzschild of mass m > in the sensethat g ij = (cid:16) m | x | (cid:17) δ ij + σ ij where ∂ I σ ij = O ( | x | − −| I | ) for all multi-indices I , which contains a sequence of outlying stable constant mean curvature spheres Σ k ⊂ M with r (Σ k ) → ∞ and r (Σ k ) H (Σ k ) → η > . It turns out that it is possible to recover a version of Theorem 1.2 without demanding homo-geneity in the expansion of the metric if instead we impose a mild growth condition on the scalarcurvature.
OTIS CHODOSH AND MICHAEL EICHMAIR
Theorem 1.4.
Let ( M, g ) be a complete Riemannian -manifold that is C -asymptotic to Schwarz-schild in the sense that g ij = (cid:16) m | x | (cid:17) δ ij + σ ij where ∂ I σ ij = O ( | x | − −| I | ) for all multi-indices I of length | I | ≤ . We also assume that either R = o ( | x | − ) as | x | → ∞ or x i ∂ i ( | x | R ) ≤ o ( | x | − ) as | x | → ∞ . (5) There does not exist a sequence of outlying stable constant mean curvature surfaces Σ k ⊂ M whoseinner radius r (Σ k ) and mean curvature H (Σ k ) satisfy r (Σ k ) → ∞ and r (Σ k ) H (Σ k ) → η > . 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 avacuum spacetime.(ii) When the metric in the chart at infinity has the special form (3) in Theorem 1.2, then R = S + o ( | x | − ) where S = X i,j =1 (cid:0) ∂ i ∂ j T ij − ∂ i ∂ i T jj (cid:1) . Note that S is homogeneous of degree −
4. Euler’s theorem gives that (5) holds if and onlyif R ≥ − o ( | x | − ). As such, Theorem 1.4 generalizes Theorem 1.2 to the non-homogeneoussetting.It is interesting to compare (5) to condition (H3) in S. Brendle’s version of Alexandrov’s theoremfor certain warped products [1]. We remark that the example constructed in Theorem 1.3 is awarped product. We also mention that S. Ma has constructed examples of ( M, g ) that containlarge unstable constant mean curvature spheres [9]. The scalar curvature in these examples isnegative in some places; see the discussion preceding the statement of Theorem 1.1 in [9] and theproof of Lemma 4.7 therein.We now turn to the case of surfaces that are very far outlying in the sense that r (Σ k ) H (Σ k ) → ∞ . 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 “Schwarzschildcontribution” to the reduced area functional leveraged in [3] is no longer on the order of O (1), butis instead decaying. As such, it is necessary to obtain rather involved estimates for the reducedfunctional. 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 > ξ ∈ R large, wecan find closed surfaces Σ ( ξ,λ ) in the chart at infinity so that the following hold: • Σ ( ξ,λ ) bounds volume 4 πλ / g . AR-OUTLYING CMC SURFACES 5 • Σ ( ξ,λ ) is the Euclidean graph of a function u ( ξ,λ ) on S λ ( λ ξ ), i.e.Σ ( ξ,λ ) = { λ ξ + y + u ( ξ,λ ) ( x ) y/λ : x = λ ξ + y ∈ S λ ( λ ξ ) } , where sup S λ ( λ ξ ) | u ( ξ,λ ) | + λ sup S λ ( λ ξ ) |∇ u ( ξ,λ ) | + λ sup S λ ( λ ξ ) |∇ u ( ξ,λ ) | = O (1 / | ξ | ) . • u ( ξ,λ ) is orthogonal to the first spherical harmonics on S λ ( λ ξ ) with respect to the Euclideanmetric. • The mean curvature of Σ ( ξ,λ ) with respect to g viewed as a function on S λ ( λ ξ ) is therestriction of a linear function.Given a sequence of connected closed stable constant mean curvature surfaces { Σ k } ∞ k =1 with r (Σ k ) → ∞ and r (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 > ξ k ∈ R when k is sufficiently large. Note that λ k > ξ k ∈ R are both large in this case. Whether ( M, g ) admits such sequences can now be decided using thefollowing result.
Theorem 1.5.
Let ( M, g ) be a complete Riemannian -manifold that is C ℓ -asymptotic to Schwarz-schild with mass m = 2 , where ℓ ≥ is an integer. Let λ > and ξ ∈ R be large. We have area g (Σ ( ξ,λ ) ) = 4 πλ − π λ R ( λ ξ ) − π λ (∆ R )( λ ξ ) − π | ξ | − + O ( λ − | ξ | − ) + O ( | ξ | − )(6) where R is the scalar curvature of ( M, g ) . This expansion can be differentiated ℓ times with respectto 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, thenthe critical point is stable if, and only if, Σ ( ξ,λ ) is a stable constant mean curvature sphere. Thisimmediately leads to the following corollary. Corollary 1.6.
Let ( M, g ) be a complete Riemannian -manifold that is C -asymptotic to Schwarz-schild in the sense that g ij = (cid:16) m | x | (cid:17) δ ij + σ ij where ∂ I σ ij = O ( | x | − −| I | ) for all multi-indices I of length | I | ≤ . Assume that the scalar curvaturevanishes. There does not exist a sequence of connected closed stable constant mean curvaturesurfaces { Σ k } ∞ k =1 in ( M, g ) with r (Σ k ) → ∞ , area g (Σ k ) → ∞ , and r (Σ k ) H (Σ k ) → ∞ . 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 We may compute the Laplacian of scalar curvature either with respect to g or with respect to the Euclideanbackground metric in the chart at infinity. The difference may be absorbed into the error terms of the expansion. OTIS CHODOSH AND MICHAEL EICHMAIR mean curvature. S. Nardulli [10] has studied the expansion for small volumes of the isoperimetricprofile 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 -manifold that is C -asymptotic to Schwarz-schild in the sense that g ij = (cid:16) m | x | (cid:17) δ ij + σ ij where ∂ I σ ij = O ( | x | − −| I | ) for all multi-indices I of length | I | ≤ . We also assume that the scalarcurvature R of ( M, g ) is radially convex at infinity in the sense that (7) x i x j ∂ i ∂ j R ≥ outside of a compact set. There does not exist a sequence of connected closed stable constant meancurvature surfaces { Σ k } ∞ k =1 in ( M, g ) with r (Σ k ) → ∞ , area g (Σ k ) → ∞ , and r (Σ k ) H (Σ k ) → ∞ . It turns out that the hypothesis (7) is surprisingly sharp. Comparing with Theorem 1.2 orTheorem 1.4, one might be lead to conjecture that it can be weakened to(i) assuming that x i x j ∂ i ∂ j R ≥ − o ( | x | − ) as | x | → ∞ , or(ii) assuming that σ ij = T ij + o ( | x | − ) as | x | → ∞ where T ij homogeneous of order −
2, and thatthe 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 -manifold ( M, g ) with non-negative scalar curvature such that, in the chart at infinity, g ij = (1 + | x | − ) δ ij + T ij + o ( | x | − ) as | x | → ∞ along with all derivatives, where T ij is homogeneous of degree − , and which contains outlyingstable constant mean curvature spheres Σ k ⊂ M with r (Σ k ) → ∞ , area g (Σ k ) → ∞ , and r (Σ k ) H (Σ k ) → ∞ . Finally, we note that there is by now an impressive body of work on stable constant meancurvature spheres in general asymptotically flat Riemannian 3-manifolds. We refer the reader toSection 2.1 in [5] for an overview and references to results in this direction.
Acknowledgments.
We thank S. Brendle for helpful conversations. M. Eichmair has been sup-ported by the START-Project Y963-N35 of the Austrian Science Fund.2.
Proof of Theorem 1.4
The proof follows the strategy of [3], with one important difference: We do not assume here thatthe deviation of the metric from Schwarzschild is homogeneous of degree − AR-OUTLYING CMC SURFACES 7 loss of generality, we may assume that the mass m is equal to 2. Thus, g ij = (1 + | x | − ) δ ij + σ ij where ∂ I σ ij = O ( | x | − −| I | )for all multi-indices I of length | I | ≤ R \ B (0). For ξ ∈ Ω and λ > ( ξ,λ ) as in Proposition 4 of [3].Moreover, the surface Σ ( ξ,λ ) is a constant mean curvature sphere (respectively, a stable constantmean curvature sphere) if, and only if, ξ is a critical point (respectively, a stable critical point) forthe map ξ area g (Σ ( ξ,λ ) ) . The derivation of Proposition 5 in [3] carries over to givearea g (Σ ( ξ,λ ) ) = 4 πλ + π F ( | ξ | ) + F σ ( ξ, λ ) + o (1) as λ → ∞ . (8)The assumption that σ is homogeneous is neither needed nor used at this point of [3]. We recallthat F ( t ) = −
14 + 16 t log t − t + (15 t − t − ) log t + 1 t − F σ ( ξ, λ ) = 12 Z S ( ξ,λ ) tr S ( ξ,λ ) σ − λ Z B ( ξ,λ ) tr σ (9)is the contribution from σ .Here and below, unless explicitly noted otherwise, all geometric operations are with respect tothe Euclidean background metric in the chart at infinity.As in [3], given ξ ∈ R and λ >
0, we will often write S ( ξ,λ ) = S λ ( λ ξ ) and B ( ξ,λ ) = B λ ( λ ξ ) . Radial variation.
The computation of the radial derivative of (9) in Section 3 of [3] usesthe top-order homogeneity of σ that is part of their assumption repeatedly. Here, we computethis derivative in the general case, employing several integration by parts to derive a geometricexpression involving the scalar curvature on the nose .( ∇ ξ F σ )( ξ, λ ) = dds (cid:12)(cid:12)(cid:12) s =1 F σ ( s ξ, λ )= λ Z S ( ξ,λ ) tr S ( ξ,λ ) ∇ ξ σ − Z S ( ξ,λ ) (tr σ ) h ξ, ν i We write ξ = ξ ⊤ + h ξ, ν i ν .= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ ) h ξ, ν i − Z S ( ξ,λ ) (tr σ ) h ξ, ν i OTIS CHODOSH AND MICHAEL EICHMAIR + λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ξ ⊤ σ )= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ ) h ξ, ν i − Z S ( ξ,λ ) (tr σ ) h ξ, ν i + λ Z S ( ξ,λ ) ( ∇ ξ ⊤ tr σ − ( ∇ ξ ⊤ σ )( ν, ν )= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ ) h ξ, ν i − Z S ( ξ,λ ) (tr σ ) h ξ, ν i + λ Z S ( ξ,λ ) ( ∇ ξ ⊤ (tr σ − σ ( ν, ν )) + 2 σ ( ∇ ξ ⊤ ν, ν )= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ ) h ξ, ν i − Z S ( ξ,λ ) (tr σ ) h ξ, ν i + λ Z S ( ξ,λ ) ((tr σ − σ ( ν, ν ))( − div S ( ξ,λ ) ξ ⊤ ) + 2 σ ( ∇ ξ ⊤ ν, ν )= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ ) h ξ, ν i − Z S ( ξ,λ ) (tr σ ) h ξ, ν i + Z S ( ξ,λ ) (tr σ ) h ξ, ν i − σ ( ν, ν ) h ξ, ν i + σ ( ξ ⊤ , ν )= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ ) h ξ, ν i + Z S ( ξ,λ ) ( σ ( ξ, ν ) − σ ( ν, ν ) h ξ, ν i ) . We define a vector field Y = h ξ, ν i σ ( ν, · ) ♯ on S ( ξ,λ ) and computediv S ( ξ,λ ) Y = 1 λ σ ( ξ, ν ) − λ h ξ, ν i σ ( ν, ν ) + h ξ, ν i tr S ( ξ,λ ) ( ∇ · σ )( ν, · ) + 1 λ h ξ, ν i tr S ( ξ,λ ) σ. The first variation formula gives λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ( ∇ · σ )( ν, · )) h ξ, ν i = 12 Z S ( ξ,λ ) (cid:0) σ ( ν, ν ) − tr S ( ξ,λ ) σ (cid:1) h ξ, ν i − Z S ( ξ,λ ) σ ( ξ, ν ) . We insert this into the above expression, and continue. dds (cid:12)(cid:12)(cid:12) s =1 F ( s ξ, λ ) = λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ∇ ν σ − tr S ( ξ,λ ) ( ∇ · σ )( ν, · )) h ξ, ν i− Z S ( ξ,λ ) (tr σ ) h ξ, ν i − σ ( ξ, ν )We write h ξ, ν i = −| ξ | + λ − h ξ, X i in the first integrand, where X is the position field.= λ Z S ( ξ,λ ) (tr S ( ξ,λ ) ( ∇ · σ )( ν, · ) − tr S ( ξ,λ ) ∇ ν σ )( | ξ | − λ − h ξ, X i ) − Z S ( ξ,λ ) ((tr σ ) h ξ, ν i − σ ( ξ, ν )) AR-OUTLYING CMC SURFACES 9 = λ Z S ( ξ,λ ) (tr ( ∇ · σ )( ν, · ) − tr ∇ ν σ )( | ξ | − λ − h ξ, X i ) − Z S ( ξ,λ ) (tr σ ) h ξ, ν i − σ ( ξ, ν )We define a vector field W = div σ − ∇ tr σ .= 12 Z S ( ξ,λ ) h ξ, λ ξ − X i h W, ν i− Z S ( ξ,λ ) (tr σ ) h ξ, ν i − σ ( ξ, ν )= 12 Z B ( ξ,λ ) div( h ξ, λ ξ − X i W ) − Z S ( ξ,λ ) ((tr σ ) h ξ, ν i − σ ( ξ, ν ))= 12 Z B ( ξ,λ ) (div W ) h ξ, λ ξ − X i− Z B ( ξ,λ ) h ξ, W i− Z S ( ξ,λ ) (tr σ ) h ξ, ν i − σ ( ξ, ν )Note that h ξ, W i = div( σ ( ξ, · ) − (tr σ ) ξ ). We apply the divergence theorem.= 12 Z B ( ξ,λ ) (div W ) h ξ, λ ξ − X i− Z S ( ξ,λ ) σ ( ξ, ν ) − (tr σ ) h ξ, ν i− Z S ( ξ,λ ) (tr σ ) h ξ, ν i − σ ( ξ, ν )= 12 Z B ( ξ,λ ) (div W ) h ξ, λ ξ − X i . Note that div W = R + O ( | x | − )where R is the scalar curvature of g . In conclusion, we obtain( ∇ ξ F σ )( ξ, λ ) = dds (cid:12)(cid:12)(cid:12) s =1 F ( s ξ, λ ) = 12 Z B ( ξ,λ ) h ξ, λ ξ − X i R + o (1) as λ → ∞ . (10)This computation connects the radial derivative of F σ with the scalar curvature R of g . Weemphasize again that our derivation parallels the proof of Proposition 7 in [3], though we do notassume the top order homogeneity of σ . Radial variation in spherical coordinates.
Assume first that R ≥ x i ∂ i ( | x | R ) ≤ . For definiteness, we assume that ξ = | ξ | e where | ξ | >
1. In this subsection, we compute the radial variation Z B λ ( λ ξ ) h λ ξ − X, ξ i R in spherical ( ρ, φ, θ ) ( ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ ) . on the complement of the z -axis. The radial line in direction(sin φ cos θ, sin φ sin θ, cos φ )intersects the sphere B λ ( λ ξ ) in the ρ -interval whose endpoints are the solutions ρ ± = λ | ξ | (cid:0) cos φ ± (1 / | ξ | − sin φ ) / (cid:1) of the quadratic equation ρ − ρ λ | ξ | cos φ + λ ( | ξ | −
1) = 0 . The intersection is non-empty for angles φ ∈ [0 , φ + ] where φ + ∈ (0 , π ) solvessin φ + = 1 / | ξ | . We then have that Z B λ ( λ ξ ) h λ ξ − X, ξ i R = Z π Z φ + Z ρ + ρ − R ( ρ, φ, θ ) (cid:0) λ | ξ | − ρ | ξ | cos φ (cid:1) ρ sin φ dρ dφ dθ = | ξ | Z π Z φ + Z ρ + ρ − ρ R ( ρ, φ, θ ) ( λ | ξ | − ρ cos φ ) sin φ dρ dφ dθ ≥ | ξ | Z π Z φ + ( | ξ | / cos φ ) R ( | ξ | / cos φ, φ, θ ) (cid:18)Z ρ + ρ − ( λ | ξ | − ρ cos φ ) dρ (cid:19) sin φ dφ dθ. Now, for every φ ∈ (0 , φ + ), Z ρ + ρ − ( λ | ξ | − ρ cos φ ) dρ = ( ρ + − ρ − ) λ | ξ | sin φ > Z B λ ( λ ξ ) h λ ξ − X, ξ i R ≥ . 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 | − ) from asymptotic flatness, we obtain upon integrating inwards from infinity AR-OUTLYING CMC SURFACES 11 that R ≥ − o ( | x | − ) as | x | → ∞ . Under these assumptions, the preceding computation leads to the estimate Z B λ ( λ ξ ) h λ ξ − X, ξ i R ≥ − o (1) as λ → ∞ . We also mention that (5) is implied by the assumption R ≥ − o ( | x | − ) and 4 R + x i ∂ i R ≤ o ( | x | − )both as | x | → ∞ . In particular, it follows from the assumptions in Theorem 1.2.3. Proof of Theorem 1.3
Our strategy here parallels the proof of Theorem 1 in [3] in that we construct our metric to havea 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 alsosimplifies the analysis) and, more importantly, they have non-negative scalar curvature.Let S : (0 , ∞ ) → ( −∞ ,
0] be a smooth function with S ( ℓ ) ( r ) = O ( r − − ℓ ) . We define a smooth function ϕ : (0 , ∞ ) → R by ϕ ( r ) = 1 r Z ∞ r ( ρ − r ) ρ S ( ρ ) dρ. Note that ϕ ′ ( r ) = − r Z ∞ r ρ S ( ρ ) dρ so(11) ( r ϕ ′ ) ′ /r = S ( r ) . Lemma 3.1.
We have that ϕ ( ℓ ) ( r ) = O ( r − − ℓ ) . Proof.
Because S ( r ) = O ( r − ), we see that ϕ ( r ) = O ( r − ) and ϕ ′ ( r ) = O ( r − ) . Using (11), we find ϕ ′′ ( r ) + 2 ϕ ′ ( r ) /r = S ( r ) . From this, the asserted decay of the higher derivatives can be verified by induction. (cid:3) On R \ { } , we define a conformally flat Riemannian metric g = (1 + 1 /r + ϕ ( r )) ¯ g = (1 + 1 /r ) ¯ g + O (1 /r ) where r = | x | . Note that g is smoothly asymptotic to Schwarzschild with mass 2. Its scalarcurvature is easily computed as R = − /r + ϕ ( r )) − ( r ϕ ′ ) ′ /r = − O (1 /r )) S ( r ) . In particular, it is non-negative on the complement of a compact set. We now make a particularchoice for S . Fix χ ∈ C ∞ ( R ) that is positive on (3 ,
4) and suppored in [3 , S ( r ) = − A ∞ X k =0 − k χ (10 − k r )where A > g (Σ ( ξ,λ ) ) = 4 πλ + 2 πF ( | ξ | ) + 12 π F σ ( ξ, λ ) + o (1) as λ → ∞ . We choose ξ ∈ R with 2 ≤ | ξ | ≤ λ = 10 j where j ≥ dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ ( sξ,λ ) ) = 2 π | ξ | F ′ ( | ξ | ) + 14 π Z X ∈ B λ ( λ ξ ) R ( X ) h ξ, λ ξ − X i + o (1)= 2 π | ξ | F ′ ( | ξ | ) − π Z X ∈ B λ ( λ ξ ) S ( | X | ) h ξ, λ ξ − X i + o (1)= 2 π | ξ | F ′ ( | ξ | ) + 2 Aπ Z Y ∈ B ( ξ ) χ ( | Y | ) h ξ, ξ − Y i + o (1) as λ → ∞ . When | ξ | = 2 √
2, the integral on the last line is negative. We choose
A > | ξ | = 5, the second term vanishes while the first term isstrictly positive. Thus, for j ≥ dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ ( sξ,λ ) )is negative when | ξ | = 2 √ | ξ | = 5. Using that the metric g is rotationallysymmetric, we see that the map ξ area g (Σ ( ξ, j ) )has a stable critical point (a local minimum) at some ξ j ∈ R with | ξ j | ∈ (2 √ , ( ξ j , 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 thework by F. Pacard and X. Xu in [11], every rotationally symmetric Riemannian manifold whosescalar curvature has a strict local extremum contains small stable constant mean curvature spheres.4.
Proof of Theorem 1.5
Consider g ij = (1 + | x | − ) δ ij + σ ij with ∂ I σ ij = O ( | x | − −| I | ) as | x | → ∞ AR-OUTLYING CMC SURFACES 13 for all multi-indices I of length | I | ≤ ξ → ∞ .For a useful analysis in this regime, it is necessary to develop the expansion of the reduced areafunctional to a higher order than was necessary in [3], which turns out to be quite delicate. Ourcomputations are also related and in part inspired by those for exact Schwarzschild in Appendix Aof [2].We also note that part of our expansion for the reduced area functional area g (Σ ( ξ,λ ) ) follows,upon rescaling the chart at infinity by λ | ξ | , from the work of S. Nardulli [10] or F. Pacard andX. Xu [11]. The estimate for the error term in (6) in e.g. [11] is O ( λ | ξ | − ) where we obtain O ( λ − | ξ | − ) + O ( | ξ | − ). Our stronger estimate is crucial for our applications here.Let ξ ∈ R and λ > r > r ∼ λ and a smooth function u ( ξ,λ ) on thesphere S r ( λ ξ ) that is perpendicular to constants and linear functions with respect to the Euclideanmetric 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 andsuch that vol g (Σ ( ξ,λ ) ) = 4 πλ / . Moreover, sup S r ( λ ξ ) | u ( ξ,λ ) | + λ sup S r ( λ ξ ) |∇ u ( ξ,λ ) | + λ sup S r ( λ ξ ) |∇ u ( ξ,λ ) | = O (1 / | ξ | ) . (12)This is a standard consequence of the implicit function theorem and elementary analysis; cf. Propo-sition 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.4.1. Estimating vol g ( B r ( a )) . Recall the following expansion for the determinant of a matrix p det( I + A ) = 1 + 12 tr A + 18 (tr A ) −
14 tr A + O ( | A | ) . Thus, we have (1 + | x | − ) q det( δ ij + (1 + | x | − ) − σ ij )= (1 + | x | − ) + 12 (1 + | x | − ) tr σ + 14 (1 + | x | − ) − (cid:18)
12 (tr σ ) − | σ | (cid:19) + O ( | x | − ) . Repeating the computations in Proposition 17 of [2] (noting the dependence of the error on r ), wefind Z B r ( a ) (1 + | x | − ) = 4 π r (1 + | a | − ) (cid:16) | a | − ) − r | a | + 97 r | a | (cid:17) + O ( r | a | − )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 istaken and then the quantity is evaluated at a ).First, note that for y ∈ B r (0) with x = a + y ,(1 + | x | − ) = (1 + | a | − ) + 2(1 + | a | − )( | a + y | − − | a | − ) + ( | a + y | − − | a | − ) | {z } = O ( | a | − | y | ) as well as | y + a | − − | a | − = − h a, y i| a | − | a | | y | − h a, y i | a | + O ( r | a | − ) . Finally, we have tr σ = tr σ + ∇ y (tr σ ) + 12 ∇ y,y (tr σ )+ 16 ∇ y,y,y (tr σ ) + 124 ∇ y,y,y,y (tr σ )+ O ( | y | | x | − ) . We will frequently consider such Taylor expansions for expressions involving σ .Combining the above expansions and using the expressions found in Appendix A, we have12 Z B r ( a ) (1 + | x | − ) tr σ = 12 (1 + | a | − ) Z B r ( a ) tr σ + (1 + | a | − ) Z B r ( a ) ( | a + y | − − | a | − ) tr σ + O ( r | a | − )= 12 (1 + | a | − ) Z B r tr σ + 14 (1 + | a | − ) Z B r ∇ y,y tr σ + 148 (1 + | a | − ) Z B r ∇ y,y,y,y tr σ − (1 + | a | − ) | a | − Z B r h a, y i ∇ y tr σ + O ( r | a | − ) + O ( r | a | − )= 2 π r (1 + | a | − ) tr σ AR-OUTLYING CMC SURFACES 15 + π r (1 + | a | − ) ∆(tr σ )+ π r ∆(∆(tr σ )) − π
15 (1 + | a | − ) r | a | − ∇ a (tr σ )+ O ( r | a | − ) + O ( r | a | − )Continuing on, we have that14 Z B r ( a ) (1 + | x | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) = π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + O ( r | a | − )Now, putting these terms together, we find thatvol g ( B r ( a )) = 4 π r (1 + | a | − ) (cid:16) | a | − ) − r | a | + 97 r | a | (cid:17) + 2 π r (1 + | a | − ) tr σ + π r (1 + | a | − ) ∆(tr σ )+ π r ∆(∆(tr σ )) − π
15 (1 + | a | − ) r | a | − ∇ a (tr σ )+ π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + O ( r | a | − ) + O ( r | a | − ) . Estimating area g ( S r ( a )) . Using the above expansion, we have that the volume form of S r ( a )becomes dµ g = (1 + | x | − ) p det( δ | S + (1 + | x | − ) − σ | S )= (1 + | x | − ) + 12 tr S σ + 14 (1 + | x | − ) − (cid:16)
12 (tr S σ ) − | σ | S | (cid:17) + O ( | a | − )= (1 + | x | − ) + 12 tr σ − r − σ ( y, y )+ 14 (1 + | x | − ) − (cid:16)
12 (tr σ ) − r − (tr σ ) σ ( y, y ) − | σ | + 2 r − | σ ( y, · ) | − r − σ ( y, y ) (cid:17) + O ( | a | − ) . As in Proposition 17 of [2], we have that Z S r ( a ) (1 + | x | − ) = 4 πr (1 + | a | − ) (cid:16) | a | − ) − r | a | + 65 r | a | (cid:17) + O ( r | a | − )We compute, using Appendix A,12 Z S r ( a ) tr σ = 12 Z S r tr σ + 14 Z S r ∇ y,y tr σ + 148 Z S r ∇ y,y,y,y tr σ + O ( r | a | − )= 2 πr tr σ + π r ∆(tr σ )+ π r ∆(∆(tr σ ))+ O ( r | a | − )and 12 Z S r ( a ) r − σ ( y, y ) = 12 Z S r r − σ ( y, y )+ 14 Z S r r − ∇ y,y σ ( y, y )+ 148 Z S r r − ∇ y,y,y,y σ ( y, y )+ O ( r | a | − )= 2 π r tr σ + π r ∆(tr σ ) + 2 π r div(div( σ ))+ π r ∆(∆(tr σ )) + π
105 ∆(div(div( σ ))+ O ( r | a | − ) . Putting these two expressions together, we find12 Z S r ( a ) tr σ − Z S r ( a ) r − σ ( y, y ) = 2 πr tr σ − π r tr σ + π r ∆(tr σ ) − π r ∆(tr σ ) − π r div(div( σ )) AR-OUTLYING CMC SURFACES 17 + π r ∆(∆(tr σ )) − π r ∆(∆(tr σ )) − π
105 ∆(div(div( σ ))+ O ( r | a | − )= 4 π r tr σ + 4 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π
105 ∆(div(div( σ ))+ O ( r | a | − ) . Finally, we compute14 Z S r ( a ) (1 + | x | − ) − (cid:16)
12 (tr σ ) − r − (tr σ ) σ ( y, y ) − | σ | + 2 r − | σ ( y, · ) | − r − σ ( y, y ) (cid:17) = 18 (1 + | a | − ) − Z S r (tr σ ) −
14 (1 + | a | − ) − Z S r r − (tr σ ) σ ( y, y ) −
14 (1 + | a | − ) − Z S r | σ | + 12 (1 + | a | − ) − Z S r r − | σ ( y, · ) | −
18 (1 + | a | − ) − Z S r r − σ ( y, y ) + O ( r | a | − )= π r (1 + | a | − ) − (tr σ ) − π r (1 + | a | − ) − (tr σ ) − πr (1 + | a | − ) − | σ | + 2 π r (1 + | a | − ) − | σ | − π r (1 + | a | − ) − (tr σ ) − π r (1 + | a | − ) − | σ | + O ( r | a | − )= 2 π r (1 + | a | − ) − (tr σ ) − π r (1 + | a | − ) − | σ | + O ( r | a | − )= − π r (1 + | a | − ) − | ˚ σ | + O ( r | a | − ) . Thus, putting this together, we find thatarea g ( S r ( a )) = 4 πr (1 + | a | − ) (cid:16) | a | − ) − r | a | + 65 r | a | (cid:17) + 4 π r tr σ + 4 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | + O ( r | a | − ) + O ( r | a | − )4.3. Estimating F ( S r ( a )) . We define F ( S r ( a )) = area g ( S r ( a )) − r − (1 + | a | − ) − vol g ( S r ( a )) . We then compute F ( S r ( a )) = 4 πr (1 + | a | − ) (cid:16) | a | − ) − r | a | + 65 r | a | (cid:17) + 4 π r tr σ + 4 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | − π r (1 + | a | − ) (cid:16) | a | − ) − r | a | + 97 r | a | (cid:17) − π r tr σ − π r ∆(tr σ ) − π r ∆(∆(tr σ ))+ 8 π
15 (1 + | a | − ) − r | a | − ∇ a (tr σ ) − π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + O ( r | a | − ) + O ( r | a | − )= 4 π r (1 + | a | − ) + 48 π r | a | + 2 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | − π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) AR-OUTLYING CMC SURFACES 19 + 8 π r | a | − ∇ a (tr σ )+ O ( r | a | − ) + O ( r | a | − ) . Estimating the mean curvature of S r ( a ) . Considerˆ g ij = ¯ g ij + ˆ σ ij where ˆ σ ij = (1 + | x | − ) − σ ij . By the computation in Lemma 7.4 of [7], we haveˆ H = H − r − tr S ˆ σ + r − ˆ σ ( y, y ) − r − tr S ( ∇ · ˆ σ )( y, · ) + r −
12 tr S ∇ y ˆ σ + O ( r − | a | − )= 2 r − − r − tr ˆ σ + 2 r − ˆ σ ( y, y ) − r − div(ˆ σ )( y ) + 12 r − ∇ y tr ˆ σ + 12 r − ∇ y ˆ σ ( y, y ) + O ( r − | a | − )for the mean curvature of S r ( a ) with respect to ˆ g . We recall the decomposition a + y = x ∈ S r ( a )and that geometric quantities are computed with respect to the Euclidean background metric ¯ g unless noted otherwise. It follows that the mean curvature of S r ( a ) with respect to g is given by H g = (1 + | x | − ) − ˆ H − | x | − ) − | x | − ˆ g ( x, ˆ ν )= (1 + | x | − ) − ˆ H − r − (1 + | x | − ) − | x | − h x, y i + O ( | a | − )= 2 r − (1 + | x | − ) − − r − (1 + | x | − ) − | x | − h x, y i− r − (1 + | x | − ) − tr σ + 2 r − (1 + | x | − ) − σ ( y, y ) − r − (1 + | x | − ) − div(ˆ σ )( y ) + 12 r − (1 + | x | − ) − ∇ y tr ˆ σ + 12 r − (1 + | x | − ) − ∇ y ˆ σ ( y, y )+ O ( | a | − )= 2 r − (1 + | x | − ) − − r − (1 + | x | − ) − | x | − h x, y i− r − (1 + | a | − ) − tr σ + 2 r − (1 + | a | − ) − σ ( y, y ) − r − div( σ )( y ) + 12 r − ∇ y tr σ + 12 r − ∇ y σ ( y, y )+ O ( r | a | − ) . Computing as in Lemma 18 of [2],2 r − (1 + | x | − ) − − r − (1 + | x | − ) − | x | − h x, y i = 2 r − (cid:0) (1 + | a | − ) − − ( | a | − | y | − | a | − h a, y i ) (cid:1) + O ( r | a | − ) . Thus, H g = 2 r − (cid:0) (1 + | a | − ) − − ( | a | − | y | − | a | − h a, y i ) (cid:1) − r − (1 + | a | − ) − tr σ + 2 r − (1 + | a | − ) − σ ( y, y ) − r − div( σ )( y ) + 12 r − ∇ y tr σ + 12 r − ( ∇ y σ )( y, y ) + O ( r | a | − ) . Now, we consider the (Euclidean) projection of H g to Λ and Λ > where Λ is the space of secondeigenfunctions on S r and Λ > is the L ( S r )-orthogonal complement of Λ ⊕ Λ ⊕ Λ .proj Λ H g = − r | a | | y | − h a, y i | a | + 2 r − (1 + | a | − ) − proj Λ σ ( y, y )+ O ( r | a | − )= − − r | a | | y | − h a, y i | a | + 2 r − (1 + | a | − ) − (cid:16) σ ( y, y ) − | y | tr σ (cid:17) + O ( r | a | − ) . For the higher eigenspaces, we will be content with the estimateproj Λ > H g = O ( | a | − ) + O ( r | a | − ) . Estimates for u . Our goal here is to improve upon the initial estimate (12).Let t ∈ [0 , S r ( a ) of the function t u . The initial normalspeed with respect to g of this family can be computed as w = u g ( y/r, ν g ) . Note that w = (1 + O ( | x | − )) u up to and including second derivatives. We will give a more precise estimate later. Thus, thesecond variation of area implies that∆ S r ( a ) g w + ( | h g | g + Ric g ( ν g , ν g )) w = H g − H Σ g + O ( λ − | ξ | − )where, as before, H g is the mean curvature of S r ( a ) with respect to g . It follows that∆ S r ( a ) u + 2 r − u = H g − H Σ g + O ( λ − | ξ | − ) . Since proj Λ > ( H g − H Σ g ) = proj Λ > H g = O ( λ − | ξ | − ) + O ( λ − | ξ | − ) , we obtain thatsup S r ( λ ξ ) | u ( ξ,λ ) | + λ sup S r ( λ ξ ) |∇ u ( ξ,λ ) | + λ sup S r ( λ ξ ) |∇ u ( ξ,λ ) | = O ( λ − | ξ | − ) + O ( | ξ | − ) . This allows us to improve the coarse estimate above to∆ S r ( a ) g w + ( | h g | g + Ric g ( ν g , ν g )) w = H g − H Σ g + O ( λ − | ξ | − ) + O ( λ − | ξ | − ) . AR-OUTLYING CMC SURFACES 21
At this point, we can improve our earlier estimate for w to w = (cid:0) (1 + | x | − ) + O ( | x | − ) (cid:1) u up to and including second derivatives. Thus∆ S r ( a ) g w = (1 + | a | − ) − ∆ S r ( a ) u + O ( λ − | ξ | − ) + O ( λ − | ξ | − ) . Continuing on, we have that | h g | g = 2 r − (1 + | a | − ) − + O ( λ − | ξ | − )and Ric g ( ν g , ν g ) = O ( λ − | ξ | − ) . Putting these estimates together, we find that(1 + | a | − ) − ∆ S r ( a ) u + 2 r − (1 + | a | − ) − u = H g − H Σ g + O ( λ − | ξ | − ) + O ( λ − | ξ | − ) . Hence, ∆ S r ( a ) proj Λ u + 2 r − proj Λ u = proj Λ (∆ S r ( a ) u + 2 r − u )= (1 + | a | − ) proj Λ H g + O ( λ − | ξ | − ) + O ( λ − | ξ | − )= 2 r | a | − ) (cid:16) σ ( y, y ) − | y | tr σ (cid:17) − r | a | | y | − h a, y i | a | + O ( λ − | ξ | − ) . This implies thatproj Λ u = − r | a | − ) (cid:16) σ ( y, y ) − | y | tr σ (cid:17) + r | a | | y | − h a, y i | a | + O ( | ξ | − )together with two derivatives. Note that in particularproj Λ u = O ( λ − | ξ | − ) + O ( | ξ | − )along with two derivatives. The above expression also implies thatproj Λ > u = O ( λ − | ξ | − ) + O ( | ξ | − )with two derivatives.4.6. Estimating F (Σ) . We have that F (Σ) = F ( S r ( a )) + Z S r ( a ) ( H g − r − (1 + | a | − ) − ) w dµ g + 12 Z S r ( a ) H g ( H g − r − (1 + | a | − ) − ) w dµ g − Z S r ( a ) (∆ S r ( a ) g w + ( | h g | g + Ric g ( ν g , ν g )) w ) w dµ g + O ( λ − | ξ | − ) + O ( λ − | ξ | − ) Recall that w = g ( y/r, ν g ) = (1 + | x | − ) (cid:16) | x | − ) − r − σ ( y, y ) (cid:17) + O ( | x | − ) . We have seen above that dµ g = (1 + | x | − ) (cid:16) | x | − ) − tr σ −
12 (1 + | x | − ) − r − σ ( y, y ) (cid:17) dµ ¯ g + O ( | x | − ) . We begin with the first term. Z S r ( a ) ( H g − r − (1 + | a | − ) − ) w dµ g = Z S r ( a ) ( H g − r − (1 + | a | − ) − ) u (1 + | x | − ) + O ( λ − | ξ | − ) + O ( λ − | ξ | − )= (1 + | a | − ) Z S r ( a ) ( H g − r − (1 + | a | − ) − ) u + O ( λ − | ξ | − ) + O ( λ − | ξ | − )= − r − (1 + | a | − ) Z S r ( a ) (cid:16) | a | | y | − h a, y i | a | (cid:17) u + 2 r − Z S r ( a ) u σ ( y, y )+ O ( λ − | ξ | − ) + O ( λ − | ξ | − )= − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) + r − (1 + | a | − ) Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y ) − r − (1 + | a | − ) − Z S r (cid:16) σ ( y, y ) − | y | tr σ (cid:17) + r − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y )+ O ( λ − | ξ | − ) + O ( | ξ | − )= − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) + 2 r Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y )1 r | a | − ) Z S r (cid:16) σ ( y, y ) − | y | tr σ (cid:17) + O ( λ − | ξ | − ) + O ( | ξ | − ) . The second term satisfies12 Z S r ( a ) H g ( H g − r − (1 + | a | − ) − ) w dµ g = O ( λ − | ξ | − ) + O ( λ − | ξ | − ) . AR-OUTLYING CMC SURFACES 23
Finally, the last term satisfies − Z S r ( a ) (∆ S r ( a ) g w + ( | h g | g + Ric g ( ν g , ν g )) w ) w dµ g = −
12 (1 + | a | − ) Z S r ( a ) (∆ S r u + 2 r − u ) u + O ( λ − | ξ | − ) + O ( λ − | ξ | − )= 2 r − (1 + | a | − ) Z S r ( a ) (proj Λ u ) + O ( λ − | ξ | − ) + O ( | ξ | − )= 12 r | a | − ) Z S r (cid:16) σ ( y, y ) − | y | tr σ (cid:17) − r Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y )+ 12 Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) + O ( λ − | ξ | − ) + O ( | ξ | − ) . Putting this together, we find that F (Σ) = F ( S r ( a )) − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) + 2 r − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y ) − r − (1 + | a | − ) − Z S r (cid:16) σ ( y, y ) − | y | tr σ (cid:17) + 12 r − (1 + | a | − ) − Z S r (cid:16) σ ( y, y ) − | y | tr σ (cid:17) − r − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y )+ 12 Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) + O ( λ − | ξ | − ) + O ( | ξ | − )= F ( S r ( a )) − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) + r − Z S r (cid:16) | a | | y | − h a, y i | a | (cid:17) σ ( y, y ) − r − (1 + | a | − ) − Z S r (cid:16) σ ( y, y ) − | y | tr σ (cid:17)
24 OTIS CHODOSH AND MICHAEL EICHMAIR + O ( λ − | ξ | − ) + O ( | ξ | − )We now use the expansions given in Appendix A.3.= F ( S r ( a )) − π | ξ | − + 8 π r | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) − π r (1 + | a | − ) − | ˚ σ | + O ( λ − | ξ | − ) + O ( | ξ | − )= 4 π r (1 + | a | − ) + 48 π r | a | + 2 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | − π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + 8 π r | a | − ∇ a (tr σ ) − π | ξ | − + 8 π r | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) − π r (1 + | a | − ) − | ˚ σ | + O ( λ − | ξ | − ) + O ( | ξ | − )= 4 π r (1 + | a | − ) − π | ξ | − + 2 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | − π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + 8 π r | a | − ∇ a (tr σ )+ 8 π r | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) + O ( λ − | ξ | − ) + O ( | ξ | − ) . AR-OUTLYING CMC SURFACES 25
Using that vol g (Ω) = π λ , we obtainarea g (Σ) = 4 π r (1 + | a | − ) + 8 π λ r − (1 + | a | − ) − − π | ξ | − + 2 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | − π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + 8 π r | a | − ∇ a (tr σ )+ 8 π r | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) + O ( λ − | ξ | − ) + O ( | ξ | − ) . Estimating r . We now use the expansionvol g (Ω) = vol g ( B r ( a )) + Z S r ( a ) w dµ g + 12 Z S r ( a ) H g w dµ g + O ( λ − | ξ | − ) + O ( | ξ | − )to relate λ and r . Note that because u is orthogonal to constants and to linear functions, Z S r ( a ) w dµ g = O ( | ξ | − ) + O ( λ − | ξ | − )and 12 Z S r ( a ) H g w dµ g = O ( λ − | ξ | − ) + O ( λ | ξ | − ) . Hence, using the expression for vol g ( B r ( a )) obtained previously, we find that4 π λ = 4 π r (1 + | a | − ) + 2 π r (1 + | a | − ) tr σ + O ( λ | ξ | − )= 4 π r (1 + | a | − ) (cid:16) | a | − ) − tr σ + O ( λ − | ξ | − ) (cid:17) . It is convenient to write λ = r (1 + | a | − ) (1 + ψ )for ψ = 12 (1 + | a | − ) − tr σ + O ( λ − | ξ | − ) = O ( λ − | ξ | − )We now estimate the first line in the expansion for area g (Σ) obtained above.4 π r (1 + | a | − ) + 8 π λ r − (1 + | a | − ) − = 4 π r (1 + | a | − ) + 8 π r (1 + | a | − ) (1 + ψ )= 4 πr (1 + | a | − ) (cid:18) ψ (cid:19) = 4 πr (1 + | a | − ) (1 + ψ ) + 4 π r (1 + | a | − ) ψ + O ( r ψ )= 4 πλ + π r (1 + | a | − ) − (tr σ ) + O ( λ − | ξ | − ) . Concluding the estimate for area g (Σ) . Combining the previous two subsections, we con-clude that area g (Σ) = 4 πλ − π | ξ | − + π r (1 + | a | − ) − (tr σ ) + 2 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ )) − π r (1 + | a | − ) − | ˚ σ | − π r (1 + | a | − ) − (cid:16)
12 (tr σ ) − | σ | (cid:17) + 8 π r | a | − ∇ a (tr σ )+ 8 π r | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) + O ( λ − | ξ | − ) + O ( | ξ | − )= 4 πλ − π | ξ | − + 2 π r ∆(tr σ ) − π r div(div( σ ))+ π r ∆(∆(tr σ )) − π r ∆(div(div( σ ))+ 8 π r | a | − ∇ a (tr σ )+ 8 π r | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) + O ( λ − | ξ | − ) + O ( | ξ | − )= 4 πλ − π | ξ | − + 2 π λ (1 + | a | − ) − (∆(tr σ ) − div(div( σ )))+ π λ (∆(∆(tr σ )) − ∆(div(div( σ )))+ 8 π λ | a | − ∇ a (tr σ )+ 8 π λ | a | (cid:0) tr σ − | a | − σ ( a, a ) (cid:1) + O ( λ − | ξ | − ) + O ( | ξ | − ) . AR-OUTLYING CMC SURFACES 27
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 considerˆ g ij = ¯ g ij + ˆ σ ij where ˆ σ ij = (1 + | x | − ) − σ ij . Then, R ˆ g = div div ˆ σ − ∆ tr ˆ σ + O ( | x | − ) . Note that div ˆ σ = (1 + | x | − ) − div σ + 4(1 + | x | − ) − | x | − σ ( x, · ) . Thus, we find thatdiv div ˆ σ = (1 + | x | − ) − div div σ + 4(1 + | x | − ) − | x | − div σ ( x )+ 20(1 + | x | − ) − | x | − σ ( x, x ) − | x | − ) − | x | − σ ( x, x )+ 4(1 + | x | − ) − | x | − div σ ( x ) + 4(1 + | x | − ) − | x | − tr σ = (1 + | x | − ) − div div σ + 8 div σ ( x )+ 4(1 + | x | − ) − | x | − (tr σ − | x | − σ ( x, x ))+ O ( | x | − ) . Similarly, ∆ tr ˆ σ = ∆ (cid:0) (1 + | x | − ) − tr σ (cid:1) = (1 + | x | − ) − ∆ tr σ + 8(1 + | x | − ) − | x | − ∇ x tr σ + (tr σ ) ∆(1 + | x | − ) − = (1 + | x | − ) − ∆ tr σ + 8 | x | − ∇ x tr σ + O ( | x | − ) . Thus, we find that R ˆ g = (1 + | x | − ) − (div div σ − ∆ tr σ )+ 4 | x | − (tr σ − | x | − σ ( x, x ))+ 8 | x | − div( σ )( x ) − | x | − ∇ x tr σ + O ( | x | − ) . It follows that R = − | x | − ) − ∆ ˆ g | x | − + (1 + | x | − ) − R ˆ g = − | x | − ) − ∆ ˆ g | x | − + (1 + | x | − ) − (div(div( σ )) − ∆ tr σ ) + 4 | x | − (tr σ − | x | − σ ( x, x ))+ 8 | x | − div( σ )( x ) − | x | − ∇ x tr σ + O ( | x | − ) . Thus, it remains to estimate ∆ ˆ g | x | − . We have that p det ˆ g = q det( δ ij + ˆ σ ij ) = 1 + 12 tr ˆ σ + O ( | x | − )and ˆ g ij = δ ij − ˆ σ ij + O ( | x | − ) . Thus, ∆ ˆ g | x | − = − | x | − σ ( x, x )+ | x | − tr σ + | x | − div σ ( x ) − | x | − ∇ x tr σ + O ( | x | − ) . Thus, we find that R = (1 + | x | − ) − (div(div( σ )) − ∆ tr σ ) − | x | − (cid:0) tr σ − | x | − σ ( x, x ) (cid:1) − | x | − ∇ x tr σ + O ( | x | − ) . Similarly, ∆ R = ∆ (div(div( σ )) − ∆ tr σ )+ O ( | x | − ) . Reduced area-functional.
We finally obtain that, for ξ ∈ R and λ > ξ, λ ) area g (Σ ( ξ,λ ) ) = 4 πλ − π λ R − π λ ∆ R − π | ξ | − + O ( λ − | ξ | − ) + O ( | ξ | − )where R is the scalar curvature of ( M, g ), and where R = R ( λ ξ ) and ∆ R = (∆ R )( λ ξ ) . The Laplacian is computed with respect to the Euclidean background metric. This is (6).We also record here the first radial derivative dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ ( sξ,λ ) = − π λ | ξ | ∂ r R − π λ | ξ | ∂ r ∆ R + 48 π | ξ | − + O ( λ − | ξ | − ) + O ( | ξ | − ) AR-OUTLYING CMC SURFACES 29 = π (cid:0) − λ | ξ | ∂ r R − λ | ξ | ∂ r ∆ R + 144 | ξ | − (cid:1) (13) + O ( λ − | ξ | − ) + O ( | ξ | − ) . where the underscore indicates evaluation at λ ξ after all derivatives are taken.This completes the proof of Theorem 1.5.5. Proof of Corollary 1.7
We assume that (
M, g ) is C -asymptotically Schwarzschild in the sense that g ij = (1 + | x | − ) δ ij + σ ij , where ∂ I σ ij = O ( | x | − −| I | ) for all multi-indices I of length | I | ≤
6. We also assume that x i x j ∂ i ∂ j R ≥ x i ∂ i R ≤ R ≥ k with r (Σ k ) → ∞ , area g (Σ k ) → ∞ , and r (Σ k ) H (Σ k ) → ∞ . For k large, we may find λ > ξ ∈ R large so that Σ k = Σ ( ξ,λ ) . By Theorem 1.5, dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ ( sξ,λ ) ) = 0so that, by (13),0 = π (cid:0) − λ | ξ | ∂ r R − λ | ξ | ∂ r ∆ R + 144 | ξ | − (cid:1) + O ( λ − | ξ | − ) + O ( | ξ | − ) . It follows that ∂ r R = O ( λ − | ξ | − ) = o ( λ − | ξ | − ) . Using this and (7), we may integrate in the in the radial direction to find that for t ≥ ∂ r R )((1 + t ) λ ξ ) ≥ ∂ r R = o ( λ − | ξ | − ) . Integrating this again, we find that R ≤ o ( λ − | ξ | − ) t + R ((1 + t ) λ ξ ) ≤ O ( λ − | ξ | − )( o (1) t + (1 + t ) − ) . Choosing t judiciously we arrange for the term in parenthesis to be o (1). We have proven that R = o ( λ − | ξ | − ) . Now, considering the first variation of area g (Σ ( ξ,λ ) ) in directions orthogonal to ξ as above. Weobtain that the full derivative satisfies DR = O ( λ − | ξ | − ) . On the other hand, because ∂ r R = o ( λ − | ξ | − ), Taylor’s theorem combined with ∂ r R ≤ ∂ r R = o ( λ − | ξ | − ) . Combining this with (7), we obtain ∂ r R = o ( λ − | ξ | − ) . Similarly, combining the facts R ≥ R = o ( λ − | ξ | − ), and DR = o ( λ − | ξ | − ) with Taylor’stheorem yields D R ≥ − o ( λ − | ξ | − ) . Similarly, we find that D ∂ r R ≤ o ( λ − | ξ | − ) . Finally, since ∂ r ∆ R = ∆ ∂ r R − | ξ | − λ − ∆ R + 2 λ − | ξ | − ∂ r R + 2 λ − | ξ | − ∂ r R, we see that ∂ r ∆ R ≤ o ( λ − | ξ | − ) . Returning to the radial first variation, we see that0 ≥ λ | ξ | ∂ r R ≥ | ξ | − + λ − O ( | ξ | − ) . 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], exceptthat 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 → ∞ .Consider S : (0 , ∞ ) → ( −∞ ,
0] a smooth function with S ( ℓ ) = O ( r − − ℓ )where S ( ℓ ) is the ℓ -th derivative. We define a smooth function ϕ : (0 , ∞ ) → R by ϕ ( r ) = 1 r Z ∞ r ( ρ − r ) ρ S ( ρ ) dρ. Arguing as in Lemma 3.1, we find that ϕ ( ℓ ) ( r ) = O ( r − − ℓ ) . On the complement of a compact subset of R we define a conformally flat Riemannian metric g = (1 + 1 /r + ϕ ( r )) ¯ g = (1 + 1 /r ) ¯ g + O (1 /r ) . Note that we can write g = (1 + 1 /r ) ¯ g + T ij + o (1 /r )for T ij = 0, so this is indeed of the form asserted in Theorem 1.8. The scalar curvature satisfies R = − O (1 /r )) S ( r ) . AR-OUTLYING CMC SURFACES 31
Now, fix χ ∈ C ∞ ( R ) with support in [4 ,
6] that is positive on (4 , χ ′ (5) = − S ( r ) = − ∞ X k =0 − k χ (10 − k r ) . Note that S ( ℓ ) ( r ) = O ( r − − ℓ ), as above.Consider ξ ∈ R with | ξ | = 10 k t for t ∈ [3 , λ = 10 k , we have thatarea g (Σ (10 k ,ξ ) ) = 4 πλ − π
15 10 k R ( λ k ξ ) − π
105 10 k (∆ R )( λ k ξ ) − π | ξ | − + O (10 − k )= 4 πλ + 2 π
15 10 − k χ ( t ) − π
35 10 − k t − + O (10 − k ) . Thus, dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ (10 k ,sξ ) ) = 2 π
15 10 − k χ ′ ( t ) + 48 π
35 10 − k t − + O (10 − k ) . For t = 7, we have dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ (10 k ,sξ ) ) = 48 π
35 10 − k − + O (10 − k ) > − − k for sufficiently large k . On the other hand, for t = 5, we have dds (cid:12)(cid:12)(cid:12) s =1 area g (Σ (10 k ,sξ ) ) = − π
15 10 − k + 48 π
35 10 − k − + O (10 − k ) < − − − k . It follows that for some t k ∈ (5 ,
7) and any ξ k ∈ R with | ξ k | = 10 k t k , the surface Σ (10 k ,ξ k ) is astable constant mean curvature sphere. This completes the proof. 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) . Note that Z B r (0) ( y i ) = 13 Z B r (0) | y | = 4 π r for all i = 1 , , . Thus, for a symmetric tensor A ij on R , we have X i,j Z B r (0) A ij y i y j = 4 π r tr A. Similarly, Z B r (0) ( y i ) = 4 π r and for i = j , Z B r (0) ( y i ) ( y j ) = 4 π r For a totally symmetric tensor B ijkl on R , we have that X i,j,k,l Z B r (0) B ijkl y i y j y k y l = X i B iiii Z B r (0) ( y i ) + 3 X i = j B iijj Z B r (0) ( y i ) ( y j )
22 OTIS CHODOSH AND MICHAEL EICHMAIR = 4 π r (cid:16) X i B iiii + X i = j B iijj (cid:17) = 4 π r X i,j B iijj . A.2.
Integrals over S r (0) . Note that Z S r (0) ( y i ) = 4 π r . It follows that, for a symmetric tensor A ij on R , X i,j Z S r (0) A ij y i y j = 4 π r tr A. Similarly, Z S r (0) ( y i ) = 4 π r for all i = 1 , , , Z S r (0) ( y i ) ( y j ) = 4 π r for all i = j. Thus, for a totally symmetric tensor B ijkl on R , we have Z S r (0) B ijkl y i y j y k y l = 4 π r B iijj If B ijkl is symmetric in the first two slots and in the second two slots separately, we obtain X i,j,k,l Z S r (0) B ijkl y i y j y k y l = X i B iiii Z S r (0) ( y i ) + X i X j = i B iijj Z S r (0) ( y i ) ( y j ) + 2 X i X j = i B ijij Z S r (0) ( y i ) ( y j ) = 4 π r (cid:16) X i B iiii + X i X j = i B iijj + 2 X i = j B ijij (cid:17) = 4 π r (cid:16) X i,j B iijj + 2 X i,j B ijij (cid:17) . Finally, we have Z S r (0) ( y i ) = 4 π r for all i = 1 , , , Z S r (0) ( y i ) ( y j ) = 4 π r when i = j, Z S r (0) ( y ) ( y ) ( y ) = 4 π r . AR-OUTLYING CMC SURFACES 33
Assume now that the tensor C ijklmn on R is symmetric in the first four indices and, separately,in the last two indices. Then, X i,j,k,l,m,n Z S r (0) C ijklmn y i y j y k y l y m y n = X i C iiiiii Z S r (0) ( y i ) + 6 X i,j distinct C iijjjj Z S r (0) ( y i ) ( y j ) + X i,j distinct C iiiijj Z S r (0) ( y i ) ( y j ) + 3 X i,j,k distinct C iijjkk Z S r (0) ( y i ) ( y j ) ( y k ) + 4 X i,j distinct C iiijij Z S r (0) ( y i ) ( y j ) + 12 X i,j,k distinct C iijkjk Z S r (0) ( y i ) ( y j ) ( y k ) = 4 π r (cid:16) X i,j,k C iijjkk + 4 X i,j,k C iijkjk (cid:17) . A.3.
Some useful integrals.
The following computations needed in the proof of Theorem 1.5 arereadily verified using the identities from the previous subsection. Z S r (0) (cid:16) | a | | y | − h a, y i | a | (cid:17) = 16 π r | a | Z S r (0) (cid:16) σ ( y, y ) − | y | tr σ (cid:17)(cid:16) | a | | y | − h a, y i | a | (cid:17) = Z S r (0) σ ( y, y ) (cid:16) | a | | y | − h a, y i | a | (cid:17) = 8 π r | a | (cid:16) tr σ − | a | − σ ( a, a ) (cid:17)Z S r (0) (cid:16) σ ( y, y ) − | y | tr σ (cid:17) = 8 π r (cid:16) | σ | − (tr σ ) (cid:17) = 8 π r | ˚ σ | . References [1] Simon Brendle,
Constant mean curvature surfaces in warped product manifolds , Publ. Math. Inst. Hautes ´EtudesSci. (2013), 247–269. MR 3090261[2] Simon Brendle and Michael Eichmair,
Isoperimetric and Weingarten surfaces in the Schwarzschild manifold , J.Differential Geom. (2013), no. 3, 387–407. MR 3080487[3] , Large outlying stable constant mean curvature spheres in initial data sets , Invent. Math. (2014),no. 3, 663–682. MR 3251832[4] Alessandro Carlotto, Otis Chodosh, and Michael Eichmair,
Effective versions of the positive mass theorem ,Invent. Math. (2016), no. 3, 975–1016. MR 3573977 [5] Otis Chodosh and Michael Eichmair,
Global uniqueness of large stable CMC surfaces in asymptotically flatRiemannian -manifolds , preprint, https://arxiv.org/abs/1703.02494 (2017).[6] Michael Eichmair and Jan Metzger, On large volume preserving stable CMC surfaces in initial data sets , J.Differential Geom. (2012), no. 1, 81–102. MR 2944962[7] Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality ,J. Differential Geom. (2001), no. 3, 353–437. MR 1916951[8] Gerhard Huisken and Shing-Tung Yau, Definition of center of mass for isolated physical systems and unique foli-ations by stable spheres with constant mean curvature , Invent. Math. (1996), no. 1-3, 281–311. MR 1369419[9] Shiguang Ma,
Unstable CMC spheres and outlying CMC spheres in AF -manifolds , preprint, https://arxiv.org/abs/1507.02767 (2015).[10] Stefano Nardulli, The isoperimetric profile of a smooth Riemannian manifold for small volumes , Ann. GlobalAnal. Geom. (2009), no. 2, 111–131. MR 2529468[11] Frank Pacard and Xingwang Xu, Constant mean curvature spheres in Riemannian manifolds , Manuscripta Math. (2009), no. 3, 275–295. MR 2481045[12] Jie Qing and Gang Tian,
On the uniqueness of the foliation of spheres of constant mean curvature in asymptot-ically flat -manifolds , J. Amer. Math. Soc. (2007), no. 4, 1091–1110. MR 2328717[13] Rugang Ye, Foliation by constant mean curvature spheres , Pacific J. Math. (1991), no. 2, 381–396.MR 1084717
Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ,08544, United States
E-mail address : [email protected] Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
E-mail address ::