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LARGE ISOPERIMETRIC SURFACES IN INITIAL DATA SETS 3 1 MICHAEL EICHMAIR ANDJAN METZGER 0 2 n Abstract. Westudytheisoperimetricstructureofasymptotically flat a J Riemannian3–manifolds(M,g)thatareC0-asymptotictoSchwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an ef- 0 3 fectivevolumecomparisontheoreminSchwarzschild. Weuseittoshow that isoperimetric regions exist in (M,g) for all sufficiently large vol- ] umes, and that they are close to centered coordinate spheres. This im- G pliesthatthevolume-preservingstableconstantmeancurvaturespheres D constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all . h competing surfaces thatenclose thesame volume. This confirmsa con- t a jectureofH.Bray. Ourresultsareconsistentwiththeuniquenessresults m forvolume-preservingstableconstantmeancurvaturesurfacesininitial data sets obtained by G. Huisken and S.-T. Yau and strengthened by [ J. Qing and G. Tian. The additional hypotheses that the surfaces be 2 spherical and far out in the asymptotic region in their results are not v necessary in our work. 9 9 9 2 . 2 1. Introduction 0 1 1 : In this paper we describe completely the large isoperimetric surfaces of v asymptotically flat Riemannian 3-manifolds (M,g) that are 0-asymptotic Xi to the Schwarzschild metric of mass m > 0. Such RiemannCian manifolds r arise naturally as initial data for the time-symmetric Cauchy problem for a the Einstein equations in general relativity. For brevity we will refer to such (M,g) as initial data sets in the introduction. A special case of the singularity theorem of Hawking and Penrose asserts that the future spacetime development of time-symmetric initial data for the Einstein equations that contains a closed minimal surface is causally incomplete. As a trial for cosmic censorship, R. Penrose suggested that the area of an outermost minimal surface in an initial data set should provide a lower bound for the ADM-mass of the spacetime development of the initial Date: January 31, 2013. MichaelEichmairgratefully acknowledgesthesupportoftheNSFgrantDMS-0906038 and of the SNFgrant 2-77348-12. 1 2 MICHAELEICHMAIRANDJANMETZGER data set. The “Penrose inequality” has been established by H. Bray in [5] andbyG.HuiskenandT.Ilmanenin[26]. Weemphasizethattheoutermost minimal surface is known to be outer area-minimizing (in particular, it is strongly stable). This variational feature is of essential importance in both available proofs of the Penrose inequality. A deep relation between the existence of stable minimal surfaces in initial data sets and their ADM- mass has been recognized and exploited by R. Schoen and S.-T. Yau in [44] in their proof of the positive energy theorem. Their work has made a profoundconnection betweenthephysicalconceptofmassandthegeometry of manifolds with non-negative scalar curvature. Itis naturaltoask if other physicalpropertiesof thespacetime development of an initial data set (M,g) are captured by its geometry. Maybe they are witnessed by the existence and behavior of special surfaces in (M,g), and their behavior? The variational properties associated with constant mean curvature surfaces in (M,g) generalize the geometric properties of the horizon in a natural way. In [27], G. Huisken and S.-T. Yau showed that the asymptotic region of an initial data set (M,g) that is 4-asymptotic to Schwarzschild of mass m > 0 C inthe senseof Definition 2.2 is foliated by strictly volume –preservingstable constant mean curvatures spheres that are perturbationsof large coordinate balls. Moreover, these spheres are unique among volume-preserving stable constant mean curvature spheres in the asymptotic region that lie outside a coordinate ball of radius H−q, where H denotes their constant mean curvature and where q (1,1]. They also concluded that as the enclosed volume ∈ 2 gets larger, these surfaces become closer and closer to round spheres whose centers converge in the limit as the volume tends to infinity to the Huisken- Yau “geometric center of mass” of (M,g). See also the announcement [8, p. 14]. R. Ye [48] has an alternative approach to proving existence of such foliations. In[40], J.QingandG.Tianstrengthenedtheuniquenessresultof [27]byshowingthefollowing: everyvolume-preservingstableconstantmean curvaturesphereinaninitialdatasetthatis 4-asymptotictoSchwarzschild C of mass m > 0 that contains a certain large coordinate ball (independent of the mean curvature of the surface) belongs to this foliation. The assumption m > 0 in the results described in the preceding paragraph is necessary: the constant mean curvature surfaces of R3 are neither strictly volume-preserving stable nor unique. In view of the results in [27, 40], and loosely speaking, positive mass has the property that it centers large, out- lying volume-preserving stable constant mean curvature surfaces. Various extensions of these results that allow for weaker asymptotic conditions have been proven in [34], [23], and [31, 30]. In [22], L.-H. Huang has shown that the “geometric center of mass” of G. Huisken and S.-T. Yau coincides with other invariantly defined notions for the center of mass. LARGE ISOPERIMETRIC SURFACES CENTER 3 In his thesis [4], H. Bray started a systematic investigation of isoperimetric surfaces in initial data sets and their relationship with mass, quasi-local mass, and the Penrose inequality. He showed that the isoperimetric surfaces of Schwarzschild are exactly round centered spheres. He deduced that the large isoperimetric surfaces in initial data sets that are compact perturbations of the exact Schwarzschild metric are also round centered spheres. Furthermore, he gave a proof of the Penrose inequality under the additional assumption that there exist connected isoperimetric surfaces enclosing any given volume in (M,g). This proof builds on H. Bray’s important obser- vation that his isoperimetric Hawking mass is monotone increasing with the volume in this case. (In fact, H. Bray pointed out that the Hawk- ing mass is monotone along foliations through connected volume-preserving stable constant mean curvature spheres whose area is increasing, such as those constructed in [27, 48].) In [4, p. 44], H. Bray conjectured that the volume-preserving stable constant mean curvatures surfaces of [27, 48] are isoperimetric surfaces. The results in the present paper confirm this. Theorem 1.1. Let (M,g) be an initial data set that is 0-asymptotic to C Schwarzschild of mass m > 0 in the sense of Definition 2.2. There exists V > 0 such that for every V V the infimum in 0 0 ≥ A (V) := inf 2(∂∗Ω): Ω is a Borel set of volume V that contains g {Hg the horizon and has finite perimeter } is achieved. Every minimizer has a smooth bounded representative whose boundary consists of the horizon and a connected surface that is close to a centered coordinate sphere. In conjunction with [27], we immediately obtain the following corollary. Corollary 1.2. If the initial data set (M,g) is 4-asymptotic to Schwarz- C schild of mass m > 0 in the sense of Definition 2.2, then the boundaries of the large isoperimetric regions of Theorem 1.1 coincide with the volume- preserving stable constant mean curvature surfaces constructed in [27]. In particular, for every sufficiently large volume there exists a unique isoperimetric region in (M,g) of that volume. The boundaries of these regions foliate the complement of a bounded subset of (M,g). It follows that the isoperimetric profile A (V) of (M,g) for large volumes V g is exactly determined. This mirrors the situation in compact Riemannian manifolds whose scalar curvature assumes its maximum at a unique point p. Under those assumptions, small isoperimetric regions are known to be perturbations of geodesic balls centered at p. (This follows from [11]. See also [36, Theorem 2.2] and [38, Corollary 3.12].) 4 MICHAELEICHMAIRANDJANMETZGER G.Huiskenhas initiated aprogramwherethemassof an initialdata setand the quasi-local mass of subsets of initial data sets are studied via isoperimetric deficits from Euclidean space. One great advantage of this approach is that only very low regularity is required of the initial data set. Theorem 1.1 identifies m as the only sensible candidate for any notion of mass that is defined in terms of A (V) when the initial data set is 0-asymptotic to g C Schwarzschild of mass m > 0, cf. [4]. A result of X.-Q. Fan, Y. Shi, and L.-T. Tam [15, Corollary 2.3] subsequent to the work of G. Huisken shows thattheADMmassofaninitialdatasetthathasintegrablescalarcurvature and which is 0-asymptotic to Schwarzschild of mass m > 0 equals m. C In a sequel [13] to this paper, we generalize our main result Theorem 1.1 to arbitrary dimensions. We also show that in Corollary 1.2, it is enough to assume that (M,g) is 2-asymptotic to Schwarzschild of mass m > 0. In C Appendix H of [13] we provide an extensive overview of the portion of the literature on isoperimetric regions on Riemannian manifolds related to our results. Structure of this paper. In Section 2 we introduce the precise decay assumptions for initial data sets that we use in this paper, and we define what exactly we mean by isoperimetric and locally isoperimetric regions. In Section 3 we prove an effective volume comparison theorem for regions in initial data sets that are 0-asymptotic to Schwarzschild. In Section C 4 we review the classical results on the regularity of isoperimetric regions and behavior of minimizing sequences for the isoperimetric problem that we need in this paper. The effective volume comparison theorem is applied in Section 5 to show that isoperimetric regions exist for every sufficiently large volume in initial data sets that are 0-asymptotic to Schwarzschild, C and that these regions become close to large centered coordinate balls as their volume increases. In Section 6 we present our most general result on the behavior of isoperimetric regions in asymptotically flat initial data sets that are not assumed to be close to Schwarzschild: either such regions slide away entirely into the asymptotically flat end of the initial data set as their volume grows large, or they begin to fill up the whole initial data set. The results in this section are largely independent of the remainder of the paper. InAppendixAwecollectseveralusefullemmasregardingintegralsof polynomially decaying quantities over surfaces with quadratic area growth. In Appendix B we summarize some steps and results from H. Bray’s thesis. Appendix C contains a “friendly” proof that limits of isoperimetric regions withdivergent volumes ininitial data sets havearea-minimizing boundaries. This fact is used in the proof of Theorem 6.1. Acknowledgements. We have had helpful and enjoyable conversations with L. Rosales and B. White regarding the geometric measure theory used LARGE ISOPERIMETRIC SURFACES CENTER 5 in this paper. We are grateful to H. Bray and S. Brendle for their interest, feedback, and enthusiasm and to G. Huisken for his great encouragement, andfor sharingwith us his perspectiveon isoperimetric mass. We are grateful for the referees for their careful reading, their useful suggestions, and for pointing out to us Corollary 2.3 in [15]. 2. Definitions and notation Definition 2.1. Let m > 0. We denote by (M ,g ) the complete Rie- m m mannian manifold (R3 0 ,φ4 3 dx2), where φ = φ (x) := 1+ m, \{ } m i=1 i m m 2r r = r(x) := x2+x2+x2, and where (x ,x ,x ) are the coordinate func- 1 2 3 P 1 2 3 tions on R3. (M ,g ) is a totally geodesic spacelike slice of the Schwarz- p m m schild spacetime of mass m > 0. We refer to (M ,g ) as the Schwarzschild m m metric of mass m > 0 for brevity, to the coordinates (x ,x ,x ) as isotropic 1 2 3 coordinates on (M ,g ), and to r(x) as the isotropic radius of x M . m m m ∈ The conformal factor φ is harmonic on R3 0 . It follows that the scalar m \{ } curvature of g vanishes. The coordinate spheres x M : r(x) = m m { ∈ r Mm will be denoted by Sr. Note that Sm is a minimal surface. It } ⊂ 2 is called the horizon of (M ,g ). The inversion x m 2 x induces m m → 2 r(x)2 a reflection symmetry of (M ,g ) across the horizon. The area of the m m (cid:0) (cid:1) isotropic coordinate sphere S is equal to φ4 4πr2. Its mean curvature with r m respect to the unit normal φ−2∂ equals φ−3(1 m)2. The Hawking mass m r m − 2r r m(Σ):= (16π)−3/2 2 (Σ) 16π H2d 2 whichisdefinedonclosed Hgm − Σ Σ Hgm surfaces Σ ⊂ Mm isqequal to m(cid:0) whenRΣ = Sr. (cid:1) Definition 2.2. An initial data set (M,g) is a connected complete Rie- mannian 3-manifold, possibly with compact boundary, such that there exists a bounded open set U M with M U = R3 B(0, 1) and such that in the ⊂ \ ∼x \ 2 coordinates induced by x = (x ,x ,x ), 1 2 3 r g δ +r2 ∂ g +r3 ∂2g C where r := x2+x2+x2. | ij − ij| | k ij| | kl ij|≤ 1 2 3 q If ∂M = , we assume that ∂M is a minimal surface, and that there are 6 ∅ no compact minimal surfaces in M besides the components of ∂M. The boundary of M is called the horizon of (M,g). Given m > 0 and an integer k 0, we say that an initial data set is k-asymptotic to Schwarzschild of ≥ C mass m > 0 if k m (1) r2+l ∂l(g g ) C where (g ) = (1+ )4δ . m ij m ij ij | − |≤ 2r l=0 X 6 MICHAELEICHMAIRANDJANMETZGER Afew remarks are in order. Thedecay assumptions for initial data sets here are quite weak. In particular, the ADM-mass is not defined for such initial data sets unless a further condition, namely the integrability of the scalar curvature, is imposed. We extend r as a smooth regular function to the entire initial data set (M,g) such that r(U) [0,1), except for the case of exact Schwarzschild ⊂ (M ,g ), where we retain the convention that r(x) denotes the isotropic m m radiusintroduced justbelow Definition 2.1. We use S to denote thesurface r x M : x = r , and B to denote the region x M : x r . We r { ∈ | | } { ∈ | | ≤ } will refer to S as the centered coordinate sphere of radius r. We will not r distinguish between the end M U of M and its image R3 B(0, 1) under \ \ 2 x. By the work of W. Meeks, L. Simon, and S.-T. Yau [33] (see also the discussionin[26,Section4]),M isdiffeomorphictoR3 minusafinitenumber of open balls whose closures are disjoint. Given an initial data set (M,g), we fix a complete Riemannian manifold (Mˆ,gˆ) diffeomorphic to R3 that contains (M,g) isometrically. We say that a Borel set U Mˆ contains the horizon if Mˆ M U. If such a set U ⊂ \ ⊂ has locally finite perimeter, we denote its reduced boundary in (Mˆ,gˆ) by ∂∗U. Note that ∂∗U is supported in M, and that 2(∂∗U) = 2(∂∗U). To Hg Hgˆ lighten the notation, we write 3(U) := 3(U M) for short. Lg Lgˆ ∩ Definition 2.3. The isoperimetric area function A : [0, ) [0, ) is g ∞ → ∞ defined by A (V) := inf 2(∂∗U) :U Mˆ is a Borel containing the horizon g {Hg ⊂ and of finite perimeter with 3(U) = V . Lg } A Borel set Ω Mˆ containing the horizon and of finite perimeter such that ⊂ 3(Ω) = V and A (V)= 2(∂∗Ω) is called an isoperimetric region of (M,g) Lg g Hg of volume V. A Borel set Ω Mˆ containing the horizon and of locally finite ⊂ perimeter is called locally isoperimetric if 2(B ∂∗Ω) 2(B ∂∗U) Hg ∩ ≤ Hg ∩ whenever B Mˆ is a bounded open subset of Mˆ and U Mˆ is a Borel set ⊂ ⊂ containing the horizon and of locally finite perimeter such that 3(Ω B) = Lgˆ ∩ 3(U B) and Ω∆U ⋐ B. Lgˆ ∩ The definition of A as well as that of isoperimetric and locally isoperi- g metric regions is independent of the particular extension (Mˆ,gˆ) of (M,g). Note that A (0) = 2(∂M) and that A (V) > 2(∂M) for every V > 0. g Hg g Hg The latter assertion follows from the assumption that the boundary of M is an outermost minimal surface. Locally isoperimetric regions arise naturally as limits of isoperimetric regions whose volumes diverge. A good example LARGE ISOPERIMETRIC SURFACES CENTER 7 to keep in mind is a half-space in R3. Standard results in geometric measure theory imply that the boundary of a (locally) isoperimetric region Ω is smooth, that Ω ∂M = unless the enclosed volume 3(Ω) = 3(Ω M) is ∩ ∅ Lg Lgˆ ∩ 0, and that isoperimetric regions are compact. Indications of the proofs of these facts with precise references to the literature to assist the reader are given in Section 4 below. The inequalities in the following lemma are well-known, and we recall them for convenient reference. Lemma 2.4. Let (M,g) be an initial data set. There exists a constant γ > 0 such that 2 (2) |f|23dL3g 3 ≤ γ |∇f|dL3g for every f ∈ C1c(M). (cid:18)ZM (cid:19) ZM If the boundary of M is empty, the constant γ > 0 can be chosen such that for any bounded Borel set Ω M with finite perimeter one has that ⊂ L3g(Ω)32 ≤ γHg2(∂∗Ω). Proof. The Sobolev inequality stated here can be obtained exactly as in [44, Lemma 3.1] by combining, in a contradiction argument, the Euclidean Sobolev inequality in the form 2 3 ZR3\B(0,1)|f|32dL3δ! ≤ γ0ZR3\B(0,1)|∇f|dL3δ for all f ∈ C1c(R3) and Poincaré–type inequalities (see [29, 8.12] for the appropriate version § with critical exponent) on precompact coordinate charts. We recall (cf. [7, Theorem II.2.1]) that the isoperimetric estimate for smoothly bounded compact regions Ω follows from applying this Sobolev inequality to approx- imations of the indicator function χ by Lipschitz functions that are one on Ω Ω and that drop off to 0 linearly in the distance from Ω. The isoperimetric inequality for sets of finite perimeter is obtained by approximation through smooth sets. (cid:3) 3. Effective refinement of H. Bray’s characterization of isoperimetric surfaces in Schwarzschild In his thesis [4], H. Bray proved that large isoperimetric surfaces of compact perturbations of the Schwarzschild metric with mass m > 0 are centered coordinate spheres in isotropic coordinates. In this section, we refine H. Bray’s work to derive an effective lower bound for the isoperimetric defect of off-centered surfaces in Schwarzschild. This bound gives us enough quan- titative information to characterize large isoperimetric surfaces in manifolds 8 MICHAELEICHMAIRANDJANMETZGER that are 0-asymptotic to Schwarzschild of mass m > 0, as we will see in C Section 5. We begin with a description of the “volume-preserving” charts used by H. Bray. We refer the reader to Appendix B for an overview of related results from H. Bray’s thesis that should be noted in this context. Let α > 0. Consider the metric cone α−2ds2 +αs2gS2 on (0, ) S2. The sphere c S2 has area α4πc2 and mean curvature 2α. On∞e c×an choose { } × c c > 0 and α > 0 so that the intrinsic geometry and (constant, outward) mean curvature of the sphere c S2 with respect to this cone coincide { } × with that of the centered sphere S (with r > m) in (M ,g ). Using the r 2 m m remarks below Definition 2.1, we see that this requires that φ7 4m 1 c3 = r3 m = r3(1+ +O( )), 1 m/(2r) r r2 − −2 m 2 2m 1 α = φm3(1− 2r)3 = 1− 3r +O(r2). Note that α (0,1) and that α 1 as r . We emphasize that α and ∈ ր → ∞ c are uniquely determined by r. The scalar curvature of this conical metric equals 21−α3. In particular, it is positive for α (0,1). αs2 ∈ ThevolumebetweenthesphereSr of(isotropic)radiusrandthehorizonSm 2 in Schwarzschild is 4π r (1+ m)6r2dr = 4πr3(1+9m+O( 1 )). The volume m 2r 3 2r r2 2 of the (punctured) disRk (0,c] S2 in the cone metric above equals 4πc3 = × 3 4πr3(1+ 4m +O( 1 )). We denote the difference between the Schwarzschild 3 r r2 volume and the cone volume by V . Note that V = 4πr3 m + O(r) = 0 0 3 2r 4πc3 m +O(c). 3 2c FollowingH.Bray,werepresentthepartoftheSchwarzschildmetric(M ,g ) m m thatliesoutsidethecenteredsphereofisotropicradiusrintheformu−2ds2+ c ucs2gS2 on [c, ) S2 for some radial function uc. This requires that ∞ × u (c) = α and ∂u = 0, and that u satisfies a certain second order ordi- c c c c | nary differential equation (to make the scalar curvaturevanish). We remark that by Birkhoff’s theorem and the constancy of the Hawking mass along centered spheres in Schwarzschild there is a first integral for u . c Finally, letgmc := u−c2ds2+ucs2gS2 bethemetricon(0,∞)×S2 withuc(s) = α for s c and u (s) is equal to u = u (s) from the preceding paragraph c c c ≤ when s c. To summarize, we have that u is 1,1, is radial, and is such c that the≥set [c, ) S2 in the gc metric is isomCetric to the exterior of a ∞ × m round sphere S of isotropic radius r in the Schwarzschild manifold of mass r m, and such that u (s) = α for s c for some constant α, such that the c ≤ LARGE ISOPERIMETRIC SURFACES CENTER 9 boundaries c S2 and S correspond and such that the mean curvature c of c S2 f{ro}m×the inside (the conical part) matches that from the outside { }× (in Schwarzschild). AkeyfeatureofthisconstructionusedbyH.Brayisthatthevolumeelement s2ds∧dgS2 of gmc is independent of c. By definition of V0, the Schwarzschild volume between the horizon and a centered Schwarzschild sphere isometric to the sphere s S2 (with s c) in ((0, ) S2,gc ) equals 4πs3 +V . { }× ≥ ∞ × m 3 0 Thus its area equals A (4πs3 +V ), where A is the function that assigns m 3 0 m to every volume (measured relative to the horizon) the area of a centered sphere in Schwarzschild that encloses that volume. On the other hand, the area of s S2 is given explicitly by u (s)4πs2. In combination this yields c { }× the following explicit expression for u : c A (V +V ) 4πs3 m 0 u (s):= for all s c, where V := ; cf. [4, p. 34]. c (36π)13V 23 ≥ 3 It is known (and easy to verify) that 2 (∂B(0,r)) m 1 Hgm = 1 +O (36π)13L3gm(B(0,r)\B(0, m2))23 − r (cid:18)r2(cid:19) and from this that 1 m 1 3V 3 (3) A (V) = 4πR2 1 +O where R := . m − R R2 4π (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) By assumption we have that u (c) = α = 1 2m + O(1). For a fixed c − 3c c2 τ (1, ) we are interested in estimating u (τc) u (c). Note that c c ∈ ∞ − A 4π(τc)3 1+ m +O(1) m 3 2τ3c c2 u (τc) = c (cid:16) (cid:0)4π(τc)2 (cid:1)(cid:17) 2 m 1 3 m 1 = 1+ +O 1 +O 2τ3c c2 − τc c2 (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19) 2m 3 1 1 = 1 +O . − 3c 2τ − 2τ3 c2 (cid:18) (cid:19) (cid:18) (cid:19) This means that for τ (1, ) fixed and τ τ we have that 0 0 ∈ ∞ ≥ 1(τ + 1)(τ 1)22m (4) u (τc) u (c) = u (τc) α 2 − c − c c − ≥ 2 τ3 3c provided that c is sufficiently large (depending only on m and τ ). This 0 quantifies the fact from [4] that u (s) is increasing for s c; see Appendix c ≥ B. In the proof of the following lemma, we supply some additional details and in fact make a slightly different claim than [4, p. 37]: 10 MICHAELEICHMAIRANDJANMETZGER Lemma 3.1 (Cf. [4, p. 37]). Consider the conical part of the metric gc m given by α−2ds2 + αs2gS2 on (0,c) S2 where α and c are such that the outward mean curvature of c S2×with respect to gc is the same as that { }× m of a centered sphere S of area α4πc2 in Schwarzschild with mass m. Then r there exists s 0 and a smooth radial function w : (s ,c] [1, ) such 0 c 0 ≥ → ∞ that w4gc is isometric to the Schwarzschild metric interior to the mean- c m convex sphere S , and such that w (c) = 1 and ∂ w = 0. r c s c c | Proof. The scalar curvature Rgmc = 21α−sα23 of the conical part of the metric gc is strictly positive. For the conformal metric w4gc to be isometric to m c m (part) of a Schwarzschild metric, it is necessary that its scalar curvature vanishes and hence that w is a solution of the elliptic (Yamabe) equation c 8∆gc wc +Rgc wc = 0. This equation reduces to a second order ordinary − m m differentialequationifwearesolvingforradialfunctions. Hencewecansolve this equation for s close to c with initial data w (c) = 1 and ∂ w = 0. By c s c c | Birkhoff’s theorem, w4gc is isometric to (part of) a Schwarzschild metric. c m To determine the mass mˆ of this metric, we evaluate its Hawking mass on the sphere c S2. Since the initial data are chosen so that the area and { }× mean curvature of this sphere coincide with that of an umbilic constant mean curvature sphere of a Schwarzschild metric of mass m, we obtain that mˆ = m. On every connected open sub-interval of (0,c] that contains c and on which the solution w exists and is non-negative, we have that c ∆gmc wc = s12∂s(s2α2∂swc) = 18Rgmc wc ≥ 0. Integrating up and using that ∂ w = 0, it follows that ∂ w 0 on any such interval. Moreover, we s c c s c | ≤ see that w (s) is a decreasing function of s. In particular, w 1 on any c c ≥ such interval. The constancy of the Hawking mass is equivalent to the existence of a first integral for the ordinary differential equation satisfied by w . We let (s ,c] be the maximally left-extended interval of existence of c 0 the solution w . Since the metric w4gc on (s ,c] S2 is isometric to (part) c c m 0 × of a Schwarzschild metric, it follows that w as s s and that we c 0 ր ∞ ց actually obtain an isometric copy of the full spatial Schwarzschild metric that lies to the mean-concave side of S . (cid:3) r Fix an isotropic sphere S in (M ,g ), let gc be the metric on (0, ) S2 r m m m ∞ × constructedabove, andletw beasinLemma3.1,extendedby1tos c,so c that ((s , ) S2,w4gc ) is isometric to (M ,g ). We will refer to it≥as the 0 ∞ × c m m m volume-preservingchartassociated withS . Recall thattheisotropic sphere r S corresponds to the coordinate sphere c S2 in ((s , ) S2,w4gc ). Frinally, let Σbeasurfacein ((s , ) S2,{w4}g×c ) homolog0ou∞s to×thehocrizmon thatencloses thesame(relative)0v∞olum×eas cc m S2. Thereadershouldkeep { }× in mind that Σ might consist of the horizon itself (enclosing volume zero) and another surface that is the boundary of a compact set that is disjoint from the horizon. In this case the area of the horizon is counted as part of the area of Σ.