Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

Appendices

Appendix A: Integral decay estimates

Our computations in this appendix take place in the part of an initial data set (M,g) that is diffeomorphic to ℝⁿ∖B ₁(0) and where

For Corollary A.3 we require in addition that for some γ∈(0,1],

(20)

i.e. that (M,g) is $\mathcal {C}^{0}$-asymptotic to Schwarzschild of mass m>0. The proofs of the statements in this appendix are straightforward extensions of those in [22, Appendix A] to higher dimensions, and we omit them.

Lemma A.1

Let (M,g) be an initial data set. Let ρ≥1 and let Σ⊂M be a closed hypersurface such that $\mathcal {H}^{n-1}_{g}(\varSigma \cap B_{r} \setminus B_{\rho}) \leq \varTheta r^{n-1}$ for all r≥ρ. Then the estimate

holds for every p>(n−1).

Proof

The proof uses the co-area formula as in [64, p. 52]. □

Corollary A.2

Let (M,g) be an initial data set. For all ρ≥1 and γ∈(0,1] and every closed hypersurface Σ⊂M with $\mathcal {H}^{n-1}_{g}(\varSigma \cap B_{r} \setminus B_{\rho}) \leq \varTheta r^{n-1}$ for all r≥ρ one has

for every β∈(0,n−2+γ).

Lemma A.3

Let (M,g) be an initial data set for which the decay assumptions (20) hold. There is a constant C′≥1 depending only on C such that for every ρ≥1 and every bounded measurable subset Ω⊂M one has

for every α∈(1,n−1+γ).

Proof

The volume elements differ by terms O(r ^2−n−γ). □

Appendix B: Hawking-mass

Let g be a rotationally symmetric metric on $(a, b) \times \mathbb {S}^{n-1}$. Given r∈(a,b), let $A = A(r) := \mathcal {H}^{n-1}_{g}(\{r\} \times \mathbb {S}^{n-1})$ denote the area of the coordinate sphere $\{r\} \times \mathbb {S}^{n-1}$, and H=H(r) its (scalar) mean curvature, computed as the tangential divergence of the normal vector field in direction ∂ _r . Define the function

$$ m (r) := \biggl(\frac{A}{\omega_{n-1}} \biggr)^{(n-2)/(n-1)} \biggl(1 - \frac{A^{2/(n-1)} H^2}{\omega_{n-1}^{2/(n-1)} (n-1)^2} \biggr). $$

This expression appears in different but equivalent form in [35, (13)]. It is constructed so as to evaluate to the mass on the centered spheres in the Schwarzschild metric. In particular, it restricts to the usual Hawking mass in dimension n=3.

Lemma B.1

(Cf. [35, Sect. 2])

Assume that the scalar curvature of g is non-negative, and that r→A(r) is non-decreasing. Then m(r) is a non-decreasing function. If m=m(c)=m(d) for some c,d∈(a,b) with c<d, then $([c, d] \times \mathbb {S}^{n-1}, g)$ is isometric through a rotationally invariant map to $(\{x \in \mathbb {R}^{n} : c' \leq |x| \leq d'\}, (1 + \frac{m}{2 |x|^{n-2}})^{\frac{4}{n-2}} \delta_{ij})$ for some 0<c′<d′ such that $1 + \frac{m}{2 (c')^{n-2}} > 0$.

Appendix C: Standard formulae

We collect several basic facts from Riemannian geometry, for ease of reference and to set forth the sign conventions that are used throughout the paper.

We begin with our conventions for the Riemann curvature tensor. Let X,Y,Z,W be vector fields on a Riemannian manifold (M,g). Let ∇ denote the Levi-Civita connection associated with g. Then $\operatorname {Rm}(X, Y, Z, W) = g(\nabla_{X} (\nabla_{Y} Z) - \nabla_{Y} (\nabla_{X} Z) - \nabla_{[X, Y]} Z, W)$. The Ricci curvature is given by $\operatorname {Rc}(X, Y) := \text{trace}_{g} \operatorname {Rm}(\cdot, X, Y, \cdot)$. The scalar curvature is given by $\operatorname {R}:= \text{trace}_{g} \operatorname {Rc}(\cdot, \cdot)$.

Lemma C.1

(Kulkarni–Nomizu product)

Let a _ij ,b _ij be two symmetric (0,2) tensors. Then the (0,4) tensor c _ijkl :=(a⊙b) _ijkl =a _jk b _il +a _il b _jk −a _ik b _jl −a _jl b _ik has the symmetries of the Riemann curvature tensor, i.e. c _ijkl =−c _jikl and c _ijkl =c _klij . If (M,g) be a Riemannian manifold and if Rm is its Riemann curvature tensor, then $\operatorname {Rm}= \frac{\overset{\circ}{ \operatorname {Rc}} \odot g}{n-2} + \frac{\operatorname {R}g \odot g}{2 n (n-1)} + W$, where $\overset{\circ}{ \operatorname {Rc}}:= \operatorname {Rc}- \frac{\operatorname {R}}{n}g$ is the trace free part of the Ricci tensor, and where W is the Weyl curvature.

Lemma C.2

(Codazzi and Gauss equations)

Let Σ be a hypersurface in a Riemannian manifold (M,g), let p∈Σ, and let {e ₁,…,e _n−1,ν} be a local orthonormal frame of TM near p such that ν restricts to a unit normal vector field along Σ. We denote by $\bar{g}_{ij} = g(e_{i}, e_{j})$ the induced metric on Σ, and by $h_{ij} := g (\nabla_{e_{i}} \nu, e_{j})$ the components of the second fundamental form of Σ with respect to ν. Let $\bar{g}^{ij} h_{ij} = H$ be the scalar mean curvature of Σ. Then $\overline{\nabla}_{k} h_{ij} - \overline{\nabla}_{i} h_{kj} = \operatorname {Rm}_{k i \nu j}$, where $\overline{\nabla}$ denotes covariant differentiation with respect to $\overline{g}$. We have that $\overline{\operatorname {Rm}}_{ijkl} = \operatorname {Rm}_{ijkl} + \,h_{il} h_{jk} - h_{ik} h_{jl}$.

Appendix D: The geometry of the spatial Schwarzschild metric

Consider the n-dimensional spatial Schwarzschild Riemannian manifold of mass m>0,

$$\Biggl(\mathbb {R}^n \setminus \{0\}, g_m := \biggl(1 + \frac{m}{2 r^{n-2}} \biggr)^{\frac{4}{n-2}} \sum_{i=1}^n d x_i^2 \Biggr), $$

where r=|x|. Given r>0, we will denote the centered coordinate sphere {x∈ℝⁿ:|x|=r} in this coordinate system by S _r . The sphere $S_{r_{h}}$ with $r_{h}= (\frac{m}{2} )^{1/(n-2)}$ is called the horizon. We record the following properties of this geometry; our sign conventions here are those of Appendix C.

• (a) The inversion $x \to r_{h}^{2}\frac{x}{|x|^{2}}$ induces a reflection symmetry of g _m across the horizon.

• (b) The g _m -area of S _r is $\phi_{m}^{\frac{2(n-1)}{n-2}} r^{n-1} \omega_{n-1}$.

• (c) The g _m -mean curvature with respect to the unit normal in direction of ∂ _r of S _r equals $\phi_{m}^{-n/(n-2)}(1-\frac{m}{2r^{n-2}}) \frac{n-1}{r}$. The horizon $S_{r_{h}}$ is a minimal surface, and the mean curvature of the spheres S _r for r>r _h is positive.

• (d) The conformal factor $\phi_{m} := 1 + \frac{m}{2r^{n-2}}$ is harmonic with respect to the Euclidean metric $\sum_{i=1}^{n} d x_{i}^{2}$. The scalar and the Weyl curvature of g _m vanish.

• (e)
  $$\operatorname {Rc}_{g_m} = \frac{(n-2) m}{r^n \phi_m^{2n/(n-2)}}\bigl(g_m - n \phi_m^{4/(n-2)} dr \otimes dr \bigr). $$
• (f)
  $$\operatorname {Rm}_{g_m} = \frac{m}{r^n \phi_m^{2n/(n-2)}} \bigl( g_m \odot g_m - n \phi_m^{4/(n-2)} (dr \otimes dr) \odot g_m \bigr). $$

Appendix E: Regularity of isoperimetric regions and the behavior of minimizing sequences

The regularity of isoperimetric regions in complete Riemannian manifolds is that of area minimizing boundaries (see [40, 58, 59] and the references therein):

Proposition E.1

Let Ω be an isoperimetric region in (M,g). Its reduced boundary ∂ ^∗ Ω is a smooth hypersurface away from a singular set of Hausdorff dimension ≤n−8.

The following technical lemma, which is needed to check the hypotheses of Theorem 3.5, follows from explicit comparison:

Lemma E.2

(Cf. [22, Lemma 4.3])

Let (M,g) be an initial data set. There exists a constant Θ>0 so that for every isoperimetric region Ω with ${\mathcal {L}}^{n}_{g}(\varOmega) \geq 1$ one has that $\mathcal {H}^{n-1}_{g}(\partial \varOmega \cap B_{r}) \leq \varTheta r^{n-1}$ for all r≥1, and that $\mathcal {H}^{n-1}_{g} (\partial \varOmega)^{\frac{1}{n-1}} {\mathcal {L}}^{n}_{g}(\varOmega)^{- \frac{1}{n}} \leq \varTheta$.

The following proposition characterizes the behavior of minimizing sequences for the isoperimetric problem (2) in initial data sets. It is a slight refinement of [58, Theorem 2.1]:

Proposition E.3

(Cf. [22, Proposition 4.2)]

Given V>0 there exists an isoperimetric region Ω⊂M—which may be empty—and a sequence of coordinate balls B(p _i ,r _i ) with p _i →∞ and 0≤r _i →r∈[0,∞) as i→∞ such that ${\mathcal {L}}^{n}_{g}(\varOmega) + {\mathcal {L}}^{n}_{g}(B(p_{i}, r_{i})) = V$ and such that $\mathcal {H}^{n-1}_{g}(\partial \varOmega) + \mathcal {H}^{n-1}_{g}(\partial B(p_{i}, r_{i})) \to A_{g}(V)$. If r>0 and ${\mathcal {L}}^{n}_{g}(\varOmega) >0$, then the mean curvature of ∂Ω equals $\frac{n-1}{r}$.

The following lemma is standard, cf. [22, Lemma 2.4].

Lemma E.4

Let (M,g) be an initial data set. There exists a constant C>0 depending only on (M,g) such that

$$ \biggl( \int_{ M} |f|^{\frac{n}{n-1}} d { \mathcal {L}}^n_{ g} \biggr)^{\frac{n-1}{n}} \leq C \int _{ M} |d f|_g d {\mathcal {L}}^n_{ g} \quad \text{\textit{for all} } f \in \mathcal {C}_c^1(M). $$

For any bounded Borel set Ω⊂M with finite perimeter one has that

$$ {\mathcal {L}}^n_{ g} (\varOmega)^{\frac{n-1}{n}} \leq C \mathcal {H}^{n-1}_{ g}\bigl(\partial^* \varOmega\bigr). $$

Appendix F: An alternative expression for the center of mass

The following lemma is an extension of [29, Lemma 2.1]. Rather than applying a density theorem as in [29], our proof below relies on an elementary integration by parts, cf. the papers [36, 37] by S. Ma.

Lemma F.1

(Cf. [29, Lemma 2.1])

Let g _ij be a metric on ℝⁿ that is $\mathcal {C}^{2}$-asymptotic to Schwarzschild of mass m>0 and asymptotically even at rate γ∈(0,1]. Then for all c>0 and γ ₁∈(0,1] there exists c′>0 such that for all p∈ℝⁿ with $|p| \leq c r^{1-\gamma_{1}}$ and r≥1 we have that

Here, $H^{S_{r}(p)}$ denotes the mean curvature of S _r (p) with respect to g.

Proof

Throughout the proof we will sum over repeated indices. Let h _ij :=g _ij −δ _ij and let $\rho = \frac{x-p}{|x-p|}$. Then

(21)

where E is an error term with |E(x)|≤c′|x−p|³⁻²ⁿ, uniformly for p such that 2|p|≤r when r is large. This follows from a calculation exactly as in the case n=3, cf. [29, Lemma 2.1]. Moreover, we have that

(22)

We claim that for each l∈{1,…,n},

(23)

To see this, define the vector field X _(l):=(x _l −p _l )h _ij ρ _i ∂ _j and note that

$$ \int_{S_r(p)} \operatorname{div}^{S_r(p)}_\delta X_{(l)} d \mathcal {H}^{n-1}_\delta = \int_{S_r(p)} H^{S_r(p)}_\delta \delta( X, \rho) d \mathcal {H}^{n-1}_\delta. $$

Using that $H^{S_{r}(p)}_{\delta}= \frac{n-1}{r}$ and that

we obtain (23). Multiply (21) by (x _l −p _l ) and integrate over S _r (p). Using (23) we arrive at

(24)

Using (22) we see that the last term has order $O(r^{3-n-\min\{\gamma,\gamma_{1}\}})$. To analyze the first term, let

$$ Y_{(l)} := \bigl(x_l( g_{ij,i} - g_{ii,j}) - (g_{jl} - g_{ii}\delta_{lj}) \bigr)\partial_j $$

so that

$$ \mathcal {C}_l = \frac{1}{2 m (n-1) \omega_{n-1}} \lim_{R\to\infty} \int _{S_R(0)} Y_{(l)}^j \frac{x_j}{R} d \mathcal {H}_\delta^{n-1}. $$

Note that $\operatorname {div}_{\delta}Y_{(l)} = x^{l} (\operatorname {R}_{g} + \,\partial g * \partial g)$. It follows that $| \operatorname {div}_{\delta}Y_{(l)} | \leq c r^{1-n-\gamma}$ and $|\operatorname {div}_{\delta}Y_{(l)} (x) + \operatorname {div}_{\delta}Y_{(l)} (-x) | \leq c r^{-n-\gamma}$.

Let R>r+|p|. Then

(25)

Hence

(26)

where we have used the estimate for the odd part of $\operatorname {div}_{\delta}Y_{(l)}$ in the first inequality, and the estimate for $\operatorname {div}_{\delta}Y_{(l)}$ and that ${\mathcal {L}}^{n}_{\delta}(B_{r}(p) \setminus B_{r}(-p)) \leq c' r^{n-\gamma_{1}}$ in the second inequality. We emphasize that the right hand side is independent of R. Using the divergence theorem, we have that

(27)

Letting R→∞ in (27) we obtain

(28)

We claim that

(29)

To see this, define the vector field Y:=(h _ij,i −h _ii,j )∂ _j , note that $\operatorname {div}_{\delta}Y = R_{g} + \partial g * \partial g = O(r^{-n-\gamma})$, and that

The lemma follows combining (24), (28) and (29). □

Appendix G: Appoximation by spheres

Lemma G.1

(Cf. [29, Lemma 4.8] and [32, Proposition 2.1])

There exist δ,c>0 depending only on n so that the following holds: Let Σ⊂ℝⁿ be a closed hypersurface. If for some constant $\bar{H} >0$ one has that $\sup_{\varSigma} |\mathring {h}| + \sup_{\varSigma}|H - \bar{H}|\leq \delta \bar{H}$, then Σ is strictly convex, and there exist $r \in (\frac{1}{2} \frac{n-1}{\bar{H}}, 2 \frac{n-1}{\bar{H}})$, p∈ℝⁿ, and a function $v \in \mathcal {C}^{2}(S_{r}(p))$ such that $\varSigma = \{ x + v(x) \frac{x - p}{|x - p|} : x \in S_{r}(p)\}$ and

$$ \sup_{S_r(p)} r^{-1} |v| + |Dv| + r\big|D^2 v\big| \leq c r \Bigl(\sup_\varSigma |\mathring {h}| + \sup_\varSigma |H-\bar{H}|\Bigr). $$

Appendix H: Overview of results on isoperimetric regions

Our intention in this section is three-fold: First, to give a complete account of all closed Riemannian manifolds whose isoperimetric regions are fully or largely characterized; second, to describe briefly all techniques and developments in the theory of isoperimetry that appear to us relevant in the context of this paper; and third, to provide the reader with an introduction to the rich literature on this subject.

8.1 H.1 Monographs and surveys

R. Osserman’s article [44] surveys the classical literature on the isoperimetric problem. We point out in particular the discussion in Sect. 2, which highlights the difference between characterizing critical and stable critical surfaces for the isoperimetric problem and establishing a sharp isoperimetric inequality, as well as the discussion in Sect. 4 on results and conjectures related to the validity of the planar Euclidean isoperimetric inequality L ²−4πA≥0 on Riemannian surfaces with non-positive curvature. R. Osserman’s article [45] gives several effective (“Bonnesen-style”) isoperimetric inequalities on Riemannian surfaces. The proofs depend on the Gauss-Bonnet theorem and F. Fiala’s method of “interior parallels”, cf. the historical discussion in Sect. II. Section III.C contains some some extensions to higher dimension. The extensive monograph [10] by Y.D. Burago and V.A. Zalgaller emphasizes the rich connection with convex and integral geometry and contains many interesting historical references. The more recent survey articles [59] by A. Ros and [55] by M. Ritoré contain a wealth of additional material and up-to-date references.

8.2 H.2 Classical isoperimetric inequality

The sharp isoperimetric inequality in the simply connected constant curvature spaces $\mathbb {R}^{n}, \mathbb{S}^{n}$, and ℍⁿ have been established rigorously in all dimensions in a series of papers by E. Schmidt in the 1940’s, cf. the Historical Remarks 10.4 as well as Sects. 8–10 in [10]. The isoperimetric regions are exactly the geodesic balls.

8.3 H.3 The case of surfaces

The isoperimetric regions of certain rotationally symmetric surfaces have been completely characterized, using curve shortening flow [5, 68], parallel surfaces techniques [24, 46, 69], and by analysis of curves of constant geodesic curvature [11, 12, 41, 52]. The introduction of the recent article [12] by A. Cañete and M. Ritoré contains a thorough overview of these results.

M. Ritoré [53] has shown that solutions of the isoperimetric problem exist for every volume in complete Riemannian planes with non-negative curvature. Conversely, in [52], M. Ritoré gives examples of complete rotationally symmetric planes in which no optimizers for the isoperimetric problem exist for any volume.

8.4 H.4 Symmetrization techniques

We refer the reader to Sects. 1.3 and 3.2 in [59] and to Sect. 1.3 [55] for brief descriptions of the symmetrization techniques by J. Steiner and H. Schwarz [66] as well as W.-T. Hsiang and W.-Y. Hsiang [27].

In [27], W.-T. Hsiang and W.-Y. Hsiang apply their symmetrization technique to reduce the study of isoperimetric regions in ℝⁿ×ℍ^m and in ℍ^m×ℍⁿ to an ODE analysis of curves in the plane. In ℝ×ℍ², the solutions are completely characterized.

R. Pedrosa and M. Ritoré [48] have characterized the isoperimetric domains of $\mathbb {S}^{1} \times \mathbb {S}^{2}$ and $\mathbb {S}^{1} \times \mathbb{H}^{2}$ as well as of $\mathbb {S}^{1} \times \mathbb {R}^{n-1}$ when 3≤n≤8, using symmetrization as in [27] and ODE and stability analysis.

R. Pedrosa [47] has used spherical symmetrization as in [27] to show that the isoperimetric regions in $\mathbb {R}\times \mathbb{S}^{n-1}$ are connected and smooth, and either topological balls or cylindrical of the form $(a,b)\times \mathbb{S}^{n-1}$. In $\mathbb {R}\times \mathbb{S}^{2}$, the author has obtained an explicit description of the isoperimetric regions.

8.5 H.5 Small isoperimetric regions in Riemannian manifolds

D. Johnson and F. Morgan [42] have shown that isoperimetric regions of small volume in closed Riemannian manifolds are perturbations of small geodesic balls. An alternative argument that applies in dimension n=3 is given in Theorem 18 of [59]. For the relationship between small isoperimetric regions and scalar curvature we refer to the work of R. Ye [70], P. Pansu [46], O. Druet [19, 20], and S. Nardulli [43].

8.6 H.6 Classifying stable constant mean curvature surfaces

Using a particular choice of test function in the stability inequality, J.L. Barbosa, M. DoCarmo [2], and J.L. Barbosa, M. DoCarmo, and J. Eschenburg [3] have shown that in the simply connected space forms, every closed volume preserving stable constant mean curvature hypersurface is a geodesic sphere.

M. Ritoré and A. Ros [56] have characterized the isoperimetric regions in $\mathbb {RP}^{3}$ and ℝ³/S _θ , where S _θ is a subgroup of O(3) generated by a translation or a screw motion, using the stability inequality in several different ways. In [57], the same authors characterize the isoperimetric regions of most products $\mathbb{T}^{2} \times \mathbb {R}$ where $\mathbb{T}^{2}$ is a flat 2-torus. (“Most” means all those from a compact subset in the non-compact moduli space of such manifolds.) A full characterization of the isoperimetric regions of T ²×ℝ, where T is a flat torus with injectivity radius 1 and area greater than a certain ϵ>0, has been given by M. Ritoré in [51].

8.7 H.7 Isoperimetric comparison

We point out the Levy-Gromov comparison theorem for the isoperimetric profile for closed Riemannian manifolds whose Ricci curvature is bounded below by that of the sphere, cf. Theorem 19 in [59]. B. Kleiner [34] has proven a sharp isoperimetric comparison result for three dimensional Hadamard manifolds. Alternative proofs of B. Kleiner’s result have been given by M. Ritoré [54] and by F. Schulze [65]; these proofs are surveyed in [55, Sects. 3.2 and 3.3].

H. Bray [7] has characterized the isoperimetric regions homologous to the horizon in the spatial Schwarzschild manifold. His method has been extended by H. Bray and F. Morgan [6] to general spherically symmetric manifolds satisfying certain conditions. J. Corvino, A. Gerek, M. Greenberg, and B. Krummel [15] have applied the methods of [6, 7] to characterize the isoperimetric regions in the spatial Reissner–Nordstrom and Schwarzschild anti de Sitter manifolds. In [9], S. Brendle and the first author use the results in [8, 22] to characterize the isoperimetric regions in the “doubled” Schwarzschild manifold, complementing the results of H. Bray in [7].

8.8 H.8 Effective isoperimetric inequalities

In [26], N. Fusco, F. Maggi, and A. Pratelli have given an effective isoperimetric inequality (“Bonnesen-style” as coined by R. Osserman [45]) for sets in ℝⁿ. Their result is sharp in a sense that the authors make precise. Their proof is based on Schwarz-Steiner symmetrization. See also the paper [25] by A. Figalli, F. Maggi, and A. Pratelli and the paper [13] by M. Cicalese and P. Leonardi for alternative proofs based respectively on optimal transport and explicit minimization.