Total mean curvature, scalar curvature, and a variational analog of Brown-York mass
aa r X i v : . [ m a t h . DG ] O c t TOTAL MEAN CURVATURE, SCALAR CURVATURE,AND A VARIATIONAL ANALOG OF BROWN-YORKMASS
CHRISTOS MANTOULIDIS AND PENGZI MIAO
Abstract.
We study the supremum of the total mean curvatureon the boundary of compact, mean-convex 3-manifolds with non-negative scalar curvature, and a prescribed boundary metric. Weestablish an additivity property for this supremum and exhibitrigidity for maximizers assuming the supremum is attained. Whenthe boundary consists of 2-spheres, we demonstrate that the finite-ness of the supremum follows from the previous work of Shi-Tamand Wang-Yau on the quasi-local mass problem in general relativ-ity. In turn, we define a variational analog of Brown-York quasi-local mass without assuming that the boundary 2-sphere has pos-itive Gauss curvature. Introduction and statement of results
Brown and York ([1], [2]) formulated a definition of quasi-local massin general relativity by employing a Hamilton-Jacobi analysis of theEinstein-Hilbert action. Given a compact spacelike hypersurface Ω ina spacetime, assuming its boundary ∂ Ω is a 2-sphere with positiveGauss curvature, the Brown-York mass of ∂ Ω is given by(1.1) m BY ( ∂ Ω; Ω) = 18 π Z ∂ Ω ( H − H ) dσ. Here dσ is the induced area element on ∂ Ω, H is the mean curvature of ∂ Ω in Ω, and H is the mean curvature of the isometric embedding of ∂ Ω into Euclidean space, R . The existence and uniqueness of such anembedding of ∂ Ω is guaranteed when ∂ Ω has positive Gauss curvature,by the solution to Weyl’s embedding problem ([13, 14]).In [16], Shi and Tam proved the following theorem which implies thepositivity of m BY ( ∂ Ω; Ω) when Ω has nonnegative scalar curvature.
The first named author’s research was partially supported by the Ric Weiland Grad-uate Fellowship at Stanford University. The second named author’s research waspartially supported by the Simons Foundation Collaboration Grant for Mathemati-cians
Theorem 1.1 ([16]) . Let (Ω , g ) be a compact, connected, Riemann-ian 3-manifold with nonnegative scalar curvature, and with nonemptyboundary ∂ Ω . Suppose ∂ Ω has finitely many components Σ j , j =1 , . . . , k , so that each Σ j is a topological 2-sphere which has positiveGauss curvature and positive mean curvature H g,j . Then (1.2) Z Σ j H g,j dσ j ≤ Z Σ j H ,j dσ j , where dσ j denotes the induced area element on Σ j , and H ,j is the meancurvature of the isometric embedding of Σ j in R . Moreover, equalityin (1.2) holds for some Σ j if and only if ∂ Ω has a unique connectedcomponent and (Ω , g ) is isometric to a convex domain in R .Remark . Our convention for the mean curvature H is that meanconvexity is equivalent to H >
0. We will often emphasize the depen-dence of a mean curvature on the interior metric by using the metricas a subscript; e.g., H g .In this paper, we consider questions concerning the total boundarymean curvature of a general compact Riemannian 3-manifold with non-negative scalar curvature which are motivated by Theorem 1.1. As anapplication of our results, we give a variational analog of Brown-Yorkmass that is free of the positive Gauss curvature restriction on theboundary.1.1. Variational results.
We first introduce the relevant definitionsbefore stating the main results.
Definition 1.1 (Fill-ins) . Let Σ , . . . , Σ k be k ≥ closed, connected,orientable surfaces endowed with Riemannian metrics γ , . . . , γ k . De-note by F (Σ ,γ ) ,..., (Σ k ,γ k ) the set of all compact, connected Riemannian3-manifolds (Ω , g ) with boundary such that: (1) ∂ Ω , with the induced metric, is isometric to the disjoint unionof (Σ j , γ j ) , j = 1 , . . . , k , (2) ∂ Ω is mean-convex; i.e., the mean curvature vector of ∂ Ω pointsinward, and (3) R ( g ) ≥ , where R ( g ) is the scalar curvature of g .Remark . In Definition 1.1, we do not prescribe the mean curvatureon the boundary, other than to require it to be positive. For a definitionof “fill-ins” that prescribes H , we refer the reader to [7].Given the set F = F (Σ ,γ ) ,..., (Σ k ,γ k ) , we define(1.3) Λ (Σ ,γ ) ,..., (Σ k ,γ k ) := sup (cid:26) π Z ∂ Ω H g dσ | (Ω , g ) ∈ F (cid:27) . otal mean curvature, scalar curvature 3 Remark . The supremum of an empty set is conventionally −∞ .In this notation, Theorem 1.1 implies that when Σ is a 2-sphere and γ , . . . , γ k are Riemannian metrics on Σ with positive Gauss curvature,then for every fill-in (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) ,(1.4) Z Σ j H g,j dσ j ≤ π Λ (Σ ,γ j ) = Z Σ j H ,j dσ j for all j = 1 , . . . , k and, therefore,Λ (Σ ,γ ) ,..., (Σ ,γ k ) ≤ k X j =1 Λ (Σ ,γ j ) .Moreover, if equality holds in 1.4 for some j , then k = 1 and (Ω , g ) isnecessarily the unique flat fill-in ( B , δ ) ∈ F (Σ ,γ ) from Weyl’s embed-ding problem.The geometric significance of inequality (1.4) is in that its right sideis a quantity that is determined only by the induced metric on theboundary component Σ j ⊂ ∂ Ω; it is independent of the interior of themanifold, and independent of all other boundary components.
Question 1.1.
If we drop the positivity requirement on the Gauss cur-vature of the boundary 2-spheres, is there still a bound for the totalmean curvature of the boundary that is intrinsic? Can we characterizemaximizers of the total mean curvature?
Regarding the functional Λ (Σ ,γ ) ,..., (Σ k ,γ k ) , we have Theorem 1.2 (Additivity) . Let Σ , . . . , Σ k be k ≥ closed, connected,orientable surfaces endowed with Riemannian metrics γ , . . . , γ k . Forall fill-ins (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) , and for every j = 1 , . . . , k , (1.5) Z Σ j H g,j dσ j ≤ π Λ (Σ j ,γ j ) ,where dσ j denotes the area element on Σ j . Moreover, (1.6) Λ (Σ ,γ ) ,..., (Σ k ,γ k ) = k X j =1 Λ (Σ j ,γ j ) ,provided F (Σ j ,γ j ) , j = 1 , . . . , k , and F (Σ ,γ ) ,..., (Σ k ,γ k ) are all nonempty.Remark . In the course of the proof we will see that F (Σ ,γ ) ,..., (Σ k ,γ k ) is empty if and only if some F (Σ j ,γ j ) , j = 1 , . . . , k , is empty.The theorem above roughly allows us to reduce our study to k = 1boundary component. We prove this theorem by employing a cut-and-fill technique for manifolds with positive scalar curvature. Christos Mantoulidis and Pengzi Miao
Theorem 1.3 (Finiteness) . If Σ is a 2-sphere endowed with an arbi-trary Riemannian metric γ , then Λ (Σ ,γ ) < ∞ . Consequently, for all Riemannian metrics γ , . . . , γ k on a 2-sphere Σ , Λ (Σ ,γ ) ,..., (Σ ,γ k ) < ∞ . This theorem is an important ingredient of our variational analog ofthe Brown-York mass. We prove it by making use of results of Wang-Yau [18] and Shi-Tam [17].
Remark . Let Σ be a 2-sphere. It is easily seen that F (Σ ,γ ) = ∅ if themetric γ has positive Gauss curvature. It is also the case that F (Σ ,γ ) = ∅ for certain metrics γ with arbitrarily negative portions (in the L sense)of curvature; indeed, by employing the method in [9], one can show that F (Σ ,γ ) = ∅ for every metric γ on Σ with λ ( − ∆ γ + K γ ) >
0, where ∆ γ and K γ are the Laplace-Beltrami operator and the Gauss curvature of(Σ , γ ), respectively. Theorem 1.4 (Rigidity) . Let Σ , . . . , Σ k be k ≥ closed, connected,orientable surfaces endowed with Riemannian metrics γ , . . . , γ k . Ifthere exists (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) such that π Z Σ j H g,j dσ j = Λ (Σ j ,γ j ) for some j = 1 , . . . , k , then k = 1 and (Ω , g ) is isometric to a mean-convex handlebody with flat interior whose genus is that of Σ (since k = 1 ). In particular, if genus(Σ ) = 0 then (Ω , g ) is an Alexandrovembedded mean-convex ball in R .Remark . The equality above will hold true for all j = 1 , . . . , k if18 π Z ∂ Ω H g dσ = Λ (Σ ,γ ) ,..., (Σ k ,γ k ) ,i.e., if the fill-in (Ω , g ) attains the supremum Λ (Σ ,γ ) ,..., (Σ k ,γ k ) .Theorem 1.4 confirms that a disconnected boundary cannot supporta maximizing configuration, as is the case in Theorem 1.1 when theboundary consists of spheres with positive Gauss curvature. Remark . It would be interesting to know whether Theorem 1.3continues to hold if we replace the boundary 2-sphere with a surface ofhigher genus. Note that Theorems 1.2 and 1.4 do not make any genusassumptions. otal mean curvature, scalar curvature 5
Quasi-local mass.
The results above, together with the implica-tion of Theorem 1.1 on Brown-York mass, suggest a variational analogof this quasi-local mass that does not require positivity of the Gausscurvature on the boundary. We describe this in the following Theorem-Definition.
Theorem 1.5 (Definition of m (Σ; Ω)) . Given a compact, connectedRiemannian 3-manifold (Ω , g ) with nonnegative scalar curvature, anda mean-convex boundary Σ which is a topological 2-sphere, define (1.7) m (Σ; Ω) = ˚Λ (Σ ,γ ) − π Z Σ H g dσ. Here γ , dσ are the metric and the area element induced on Σ by g , and, (1.8) ˚Λ (Σ ,γ ) := sup (cid:26) π Z ∂M H h dσ | ( M, h ) ∈ ˚ F (cid:27) , where ˚ F = ˚ F (Σ ,γ ) is as in Definition 1.2 below. Then m (Σ; Ω) is (a) well-defined and finite, (b) nonnegative, i.e., m (Σ; Ω) ≥ , and (c) m (Σ; Ω) = 0 only if (Ω , g ) is a flat 3-ball immersed in R .Finally, when Σ has positive Gauss curvature, m (Σ; Ω) = m BY (Σ; Ω) .Hence, m (Σ; Ω) may be viewed as a variational analog and generaliza-tion of Brown-York mass. Definition 1.2 (Fill-ins, II) . Let Σ be a closed, connected, orientablesurface endowed with a Riemannian metric γ . Denote by ˚ F (Σ ,γ ) the setof all compact, connected Riemannian 3-manifolds (Ω , g ) with boundarysuch that: (1) ∂ Ω has a connected component Σ O which, with the induced met-ric, is isometric to (Σ , γ ) , (2) Σ O is mean-convex; i.e., the mean curvature vector of Σ O pointsinward, (3) ∂ Ω \ Σ O , if nonempty, is a minimal surface, possibly discon-nected, and (4) R ( g ) ≥ , where R ( g ) is the scalar curvature of g .Remark . Clearly F (Σ ,γ ) ⊆ ˚ F (Σ ,γ ) , and therefore Λ (Σ ,γ ) ≤ ˚Λ (Σ ,γ ) .We will, in fact, prove that these last two quantities coincide. Thus, m (Σ; Ω) will remain unchanged if one replaces ˚ F with F . We never-theless choose to use this enlarged class in the definition of m (Σ; Ω),because it is more suitable for discussion on quasi-local mass and moreconvenient for the cut-and-fill operations: Christos Mantoulidis and Pengzi Miao (1) For an element (Ω , g ) ∈ ˚ F (Σ ,γ ) , the portion ∂ Ω \ Σ O of the bound-ary, if nonempty, represents horizons of black holes detected byobservers at (the outer boundary) Σ O . For example, the regionbounded by a rotationally symmetric sphere (Σ , γ ) of positivemean curvature and the horizon in a half spatial Schwarzschildmanifold of positive mass is now a valid fill-in of (Σ , γ ), whileit wasn’t in the original class F (Σ ,γ ) .(2) Elements in ˚ F (Σ ,γ ) serve as building blocks, effectively allow-ing to deduce F (Σ ,γ ) ,..., (Σ k ,γ k ) -related results from them via acut-and-fill technique; we cut composite manifolds across min-imal surfaces, and, when necessary, fill in holes with 3-balls ofpositive scalar curvature. Remark . The motivation behind Theorem 1.5 is that only the in-tegral quantity R ∂ Ω H dσ is of actual interest for the purposes of theactual definition of Brown-York mass in (1.1), and not the pointwisequantity H . Moreover, when Σ has positive Gauss curvature, Theorem1.1 characterizes this integral quantity as Z Σ H dσ = sup ( M,h ) ∈F Z ∂M H h dσ, over an appropriate class F of fill-ins. Remark . It seems a challenging question to check whether mean-convex domains Ω ⊂ R , with Σ = ∂ Ω a 2-sphere, necessarily maximizethe total mean curvature on their boundary relative to all competitorsin F (Σ ,g R | Σ ) ; i.e., is m (Σ; Ω) = 0 for all mean-convex domains Ω ⊂ R which are bounded by a 2-sphere Σ? If Σ is strictly convex in R , then,indeed, m (Σ; Ω) = m BY (Σ; Ω) = 0 by Theorem 1.1. Organization of the paper.
In Section 2 we establish the basic cut-ting, filling, and doubling lemmas that will be used throughout thepaper. In Section 3 we establish, for ˚ F and ˚Λ, finiteness and rigidityresults in the spirit of Theorems 1.3 (finiteness) and 1.4 (rigidity). InSection 4 we first prove Theorem 1.2 (additivity) for F , Λ, and use itto derive Theorems 1.3 and 1.4 for F and Λ from the correspondingresults for ˚ F and ˚Λ. In Section 5 we prove Theorem 1.5 pertaining to m (Σ; Ω). Remark . After the first version of this paper was completed, welearned that the assertion Λ (Σ ,γ ) < ∞ in Theorem 1.3, and its proof,appear independently in a recent work by Lu [8], which establishesa priori estimates for certain isometric embeddings of 2-spheres intogeneral Riemannian 3-manifolds. otal mean curvature, scalar curvature 7 Acknowledgments.
The first named author would like to thank RickSchoen for his continued guidance and Otis Chodosh for a helpful dis-cussion in the early stages of this work. Both authors would like tothank Pengfei Guan for bringing the work in [8] to our attention, aswell as the referees for their helpful comments and their careful readingof the manuscript. 2.
Technical lemmas
In this section we prove some technical lemmas that will be invokedmultiple times throughout the paper.
Lemma 2.1 (Cutting) . Let Σ , . . . , Σ k be k ≥ closed, connected,orientable surfaces endowed with Riemannian metrics γ , . . . , γ k . If (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) , then for every j = 1 , . . . , k there exists an (Ω j , g j ) ∈ ˚ F (Σ j ,γ j ) such that (1) the mean curvature of Σ j is the same in (Ω , g ) and (Ω j , g j ) , (2) R ( g j ) > on Ω j if R ( g ) > on Ω , and (3) ∂ Ω j \ Σ j consists of stable, orientable minimal surfaces.Proof. For convenience, first suppose Ω is orientable. Write S j for theboundary component of Ω corresponding to our fixed Σ j . Since themean curvature vector points inward on ∂ Ω, standard results in geo-metric measure theory show that there exists a smooth, oriented mini-mal surface S in the interior of Ω that minimizes area in the homologyclass [ S j ] ∈ H (Ω; Z ). (Specifically, we minimize area in the class ofintegral currents homologous to S j .) Denote Ω j the metric completionof the component of Ω \ S containing S j , and g j its induced metric.Note that Ω j is orientable. Then (Ω j , g j ) ∈ ˚ F (Σ j ,γ j ) and satisfies allthree assertions.If Ω is nonorientable, consider its orientation double cover π : ˜Ω → Ω.Using the fact that S j itself is orientable, one can show that π − ( S j )is a disjoint union of two copies of S j . Denote one of them by ˜ S j .The lemma follows by repeating the previous proof with (Ω , g ) and S j replaced by ( ˜Ω , π ∗ ( g )) and ˜ S j , respectively. (cid:3) Lemma 2.2 (Filling) . Let Σ be a closed, connected, orientable surfacewith a Riemannian metric γ . Suppose (Ω , g ) ∈ ˚ F (Σ ,γ ) is such that: (1) Σ H = ∂ Ω \ Σ O is nonempty, (2) R ( g ) > on Σ H ⊂ ∂ Ω , and (3) every component of Σ H is a stable minimal 2-sphere.Then for every η > , there exists ( D, h ) ∈ F (Σ ,γ ) with H h > H g − η on its boundary S O , which corresponds to Σ O . Christos Mantoulidis and Pengzi Miao
Proof.
The first eigenvalue of the operator − ∆ T ℓ + K ℓ is strictly positive on each sphere T ℓ on the portion Σ H of ∂ Ω (cf. [5, 9]).Here ∆ T ℓ , K ℓ are the Laplacian, and Gauss curvature of T ℓ with respectto the induced metric from g . By the method in [9], we can glue 3-ballsof positive scalar curvature onto T ℓ to obtain a compact manifold D with ∂D = S O . More precisely, for every ℓ one can apply Lemma 1.3in [9] to first produce a cylinder of positive scalar curvature, and thenattach a spherical cap to it. If the resulting metric ˆ g on D were smoothacross every T ℓ , then ( D, ˆ g ) ∈ F (Σ ,γ ) . In general ˆ g will not be smoothacross the T ℓ , so we apply [11, Proposition 3.1] to ( D, ˆ g ) followed bya small conformal deformation to obtain another metric h on D suchthat ( D, h ) ∈ F (Σ ,γ ) and H S O ,h ≥ H S O , ˆ g − η/ ≥ H Σ O ,g − η . (cid:3) Lemma 2.3 (Doubling) . Let Σ be a closed, connected, orientable sur-face with a Riemannian metric γ . Suppose (Ω , g ) ∈ ˚ F (Σ ,γ ) , and that Σ H = ∂ Ω \ Σ O is nonempty. Let D denote the doubling of Ω across Σ H , so that ∂D = Σ O ∪ Σ ′ O , where Σ ′ O denotes the mirror image of Σ O .For every η > there exists a scalar-flat Riemannian metric h on D such that ( D, h ) satisfies: (1) Σ O with the induced metric from h is isometric to (Σ , γ ) , (2) H h > H g on Σ O , and (3) H ′ h > H g − η on Σ ′ O ,where H g denotes the mean curvature of Σ O in (Ω , g ) , and H h , H ′ h denote the mean curvatures of Σ O , Σ ′ O in ( D, h ) .Proof. The following argument is motivated by that used by Jaureguiin the proof of [7, Proposition 7].First, we reduce to the case of R ( g ) being identically zero. If R ( g ) ≥ R ( g ) is not identically 0, consider a conformally deformed metric˜ g = u g on Ω where u > ∆ g u − R ( g ) u = 0 on Ω u = 1 at Σ O∂u∂ν = 0 at Σ H . Here ∆ g is the Laplacian of g , and ν is the outward unit normal to ∂ Ωwith respect to g . Then H ˜ g = 0 at Σ H , and H ˜ g = H g + 4 ∂u∂ν at Σ O . Itfollows from the strong maximum principle and the fact ∂u∂ν = 0 at Σ H that the maximum of u is attained at Σ O . Hence, by the the strongmaximum principle again, ∂u∂ν > H ˜ g > H g at Σ O . At this point,we replace our (Ω , g ) with (Ω , ˜ g ). If R ( g ) were identically zero to beginwith, take ˜ g = g . otal mean curvature, scalar curvature 9 In any case, H ˜ g ≥ H g on Σ O and H ˜ g = 0 on Σ H . Given a smallconstant ε ∈ (0 , φ > φ = 1 at Σ O and φ = 1 − ε at Σ H . Let φ = (2 − ε ) − φ . Considertwo conformally deformed metrics g = φ ˜ g and g = φ ˜ g . They satisfythe following properties:(i) the induced metrics on Σ H from g and g agree,(ii) the mean curvature of Σ H in (Ω , g ) with respect to the inwardnormal agrees with the mean curvature of Σ H in (Ω , g ) withrespect to the outward normal,(iii) the mean curvature H g of Σ O in (Ω , g ) satisfies H g > H ˜ g bythe strong maximum principle, and(iv) the mean curvature H g of Σ O in (Ω , g ) remains positive andarbitrarily close to H ˜ g , say H g > H ˜ g − η/
2, if ε is small enough.Now attach (Ω , g ) and (Ω , g ) along Σ H , and call the resulting manifold( D, h ). In (
D, h ), denote Σ O coming from (Ω , g ) still by Σ O whiledenote Σ O coming from (Ω , g ) by Σ ′ O . If the metric h were smoothacross Σ H , then it satisfies all the properties required. In general,we can replace h with another metric that is obtained by applying[11, Proposition 3.1] to ( D, h ) at Σ H followed by a small conformaldeformation that fixes the boundary. The result follows. (cid:3)
3. ˚ F and ˚Λ for a prescribed surface We first prove
Proposition 3.1 (Finiteness for ˚ F ) . Let Σ be a 2-sphere. Given aRiemannian metric γ on Σ , there exists a constant C > , dependingonly on γ , such that (3.1) sup (Ω ,g ) ∈ ˚ F (Σ ,γ ) Z ∂ Ω H g dσ < C. As a result, ˚Λ (Σ ,γ ) := sup (cid:26) π Z ∂ Ω H g dσ | (Ω , g ) ∈ ˚ F (cid:27) < ∞ . To prove Proposition 3.1, we make use of the following result of Shiand Tam in [17], which is built on the work on Wang and Yau in [18].
Theorem 3.1 ([17]) . Let (Ω , g ) be a compact, orientable Riemannian3-manifold with boundary Σ , and scalar curvature R ( g ) ≥ − κ forsome constant κ > . Suppose Σ is a topological 2-sphere with Gausscurvature K > − κ and positive mean curvature H . Let ι : Σ → H − κ be an isometric embedding and denote Σ = ι (Σ) , which is a convex surface in H − κ . Let D ⊂ H − κ be the bounded region enclosed by Σ .Then for any p ∈ D , (3.2) Z Σ H cosh κr ( p, ι ( z )) dσ ( z ) ≤ Z Σ H cosh κr ( p, y ) dσ ( y ) , where r ( p, · ) denotes the distance to p in H − κ , dσ is the area elementon Σ , and H is the mean curvature of Σ in H − κ . Moreover, equalityin (3.2) holds if and only if (Ω , g ) is isometric to the domain boundedby Σ in H − κ .Remark . The existence of such an embedding ι is given by a theo-rem of Pogorelov [14]. Remark . Even though not explicitly stated, the proof of Theorem3.1 in [17] assumes Ω is orientable since the proof uses the fact that Ωis a spin manifold.As the first step toward proving Proposition 3.1, we want to point outthat Theorem 3.1 continues to hold for manifolds with more than oneboundary component under suitable boundary conditions; moreover, itholds without the orientability assumption.
Proposition 3.2.
Let (Ω , g ) be a compact, orientable Riemannian 3-manifold with boundary ∂ Ω , and scalar curvature R ( g ) ≥ − κ forsome constant κ > . Suppose (a) ∂ Ω has a connected component Σ that is a topological 2-spherewith Gauss curvature K > − κ and mean curvature H > ,and (b) ∂ Ω \ Σ , if nonempty, has mean curvature H ≥ − κ .Then the conclusion of Theorem 3.1 holds for such an (Ω , g ) .Proof. As Ω is an oriented 3-manifold, Ω is spin. In [3, Theorem 4.7],Chru´sciel and Herzlich proved that if (
M, g ) is an n -dimensional, spin,asymptotically hyperbolic manifold with a compact boundary ∂M suchthat R ( g ) ≥ − n ( n − κ and H ≥ − ( n − κ, where R ( g ) is the scalar curvature of g and H is the mean curvatureof ∂M (with respect to the outward normal), then the positive masstheorem holds on such an ( M, g ). Now going through the same proof ofTheorem 3.1 in [17] which involves carrying out the same constructionin [18], but replacing the positivity of the mass expression used in[18] by the positivity of the mass provided in [3, Theorem 4.7], oneconcludes that Theorem 3.1 holds for such an (Ω , g ). (cid:3) otal mean curvature, scalar curvature 11 Proposition 3.3.
Proposition 3.2 continues to hold without the ori-entability assumption on Ω .Proof. Let (Ω , g ) be a compact Riemannian 3-manifold satisfying allassumptions in Proposition 3.2 except that Ω is nonorientable. Let˜Ω be the orientation double cover of Ω and let π : ˜Ω → Ω be thecorresponding covering map. It is easily seen that ∂ ˜Ω = π − ( ∂ Ω) and ∂ ˜Ω doubly covers ∂ Ω. Let S = ∂ Ω \ Σ, which can be empty. Let˜ S = π − ( S ) and ˜Σ = π − (Σ). Since Σ is a 2-sphere and ˜Σ doublycovers Σ, ˜Σ is the disjoint union of two 2-spheres, which we denoteby ˜Σ (1) and ˜Σ (2) . Now let ˜ g = π ∗ g on ˜Ω, which has scalar curvature R (˜ g ) ≥ − κ . Let ˜ H denote the mean curvature of ∂ ˜Ω = ˜ S ∪ ˜Σ (1) ∪ ˜Σ (2) in ( ˜Ω , ˜ g ) (with respect to the outward normal). Then ˜ H ≥ − κ on ˜ S and ˜ H > ( i ) , i = 1 ,
2. Hence, Proposition 3.2 is applicable to( ˜Ω , ˜ g ). Therefore, the conclusion of Theorem 3.1 holds for each ˜Σ ( i ) , i = 1 ,
2, which in turn shows that the same is true for Σ in (Ω , g ). (cid:3) Proof of Proposition 3.1.
Let κ >
K > − κ where K is the Gauss curvature of γ . Let ι be an isometric embeddingof (Σ , γ ) in H − κ . Let D ⊂ H − κ be the bounded region enclosed bythe convex surface Σ = ι (Σ). Let H > in H − κ and let p ∈ D be a fixed point.Take (Ω , g ) ∈ ˚ F (Σ ,γ ) , then R ( g ) ≥ > − κ in Ω and H = 0 > − κ at ∂ Ω \ Σ O . By Proposition 3.3, the conclusion of Theorem 3.1 holdsfor (Ω , g ) at Σ O , i.e.(3.3) Z Σ O H cosh κr ( p, ι ◦ φ ( z )) dσ ( z ) ≤ Z Σ H cosh κr ( p, y ) dσ ( y ) , where φ is a given isometry between Σ O and (Σ , γ ). Let r ∗ = min { r ( p, y ) | y ∈ Σ } > , then (3.3) gives(3.4) Z Σ O Hdσ ≤ (cosh κr ∗ ) − Z Σ H cosh κr ( p, y ) dσ ( y ) . Clearly, the right-hand side of (3.4) is a constant determined only by(Σ , γ ). This proves Proposition 3.1. (cid:3)
Remark . In Proposition 3.1, the assump-tion H g > O for an (Ω , g ) ∈ ˚ F (Σ ,γ ) can be relaxed to H g ≥ , g ) satisfies all the assumptions imposed on an“fill-in” in Definition 1.2, except that Σ O is weakly mean-convex, i.e., H g ≥ O . Let φ ≥ w be the unique solution to(3.5) (cid:26) ∆ g w − R ( g ) w = φ on Ω w = 0 at ∂ Ω . Then w ≤ ∂w∂ν > ∂ Ω by the strong maximum principle,where ν is the outward unit normal to ∂ Ω in (Ω , g ). Given a small ε >
0, consider g ε = (1 + εw ) g . Then H g ε = H g + 4 ε ∂w∂ν > O , H g ε = 4 ε ∂w∂ν > ∂ Ω \ Σ O , and(3.6) R ( g ε ) = − (1 + εw ) − [ ε (8∆ g w − R ( g ) w ) − R ( g )] . Now we repeat the proof of Proposition 3.1, with g replaced by g ε . Notethat the assumptions R ( g ) ≥ H = 0 at ∂ Ω \ Σ O in the proof ofProposition 3.1 are only used to yield R ( g ) > − κ and H > − κ at ∂ Ω \ Σ O . On the other hand, if ε is sufficiently small, R ( g ε ) > − κ by (3.6) and the fact R ( g ) ≥
0. We already know H g ε > ∂ Ω \ Σ O .Therefore, the same proof leading to (3.4) gives(3.7) Z Σ O H g ε dσ < C where C is the quantity that is on the right-side of (3.4). As H g < H g ε ,we conclude (3.7) holds with H g ε replaced by H g . Thus, the claim inthis remark follows.Next, we want to prove Proposition 3.4 (Rigidity in ˚ F ) . Let Σ be an arbitrary closed, con-nected, orientable surface, endowed with a Riemannian metric γ . Ifthere exists (Ω , g ) ∈ ˚ F (Σ ,γ ) attaining the supremum ˚Λ (Σ ,γ ) , i.e., if π Z ∂ Ω H g dσ = ˚Λ (Σ ,γ ) ,then (Ω , g ) is isometric to a mean-convex handlebody with flat interior,whose genus is that of Σ . In particular, if genus(Σ ) = 0 then (Ω , g ) isan Alexandrov embedded mean-convex ball in R . One step in proving Proposition 3.4 is to exclude the possibility ofhaving minimal boundary components appear on a maximizing fill-in.We do this by using minimal boundary components as a tool that helpsus increase the mean curvature of Σ O . Lemma 3.1.
Let (Ω , g ) ∈ ˚ F (Σ ,γ ) . If ∂ Ω \ Σ O = ∅ , then π Z ∂ Ω H g dσ < ˚Λ (Σ ,γ ) . otal mean curvature, scalar curvature 13 Proof.
Employ Lemma 2.3 to double (Ω , g ) across Σ H = ∂ Ω \ Σ O if itis not empty. For small η >
0, the resulting manifold (
D, h ) is going tohave the mean curvature H h of Σ O in ( D, h ) satisfying H h > H g andthe mean curvature H ′ h of Σ ′ O in ( D, h ) satisfying H ′ h >
0; here Σ ′ O isthe mirror image of Σ O in D . The metric on Σ O is still γ , while themetric on Σ ′ O may have changed to some γ ′ . Cut ( D, h ) via Lemma2.1 to isolate the boundary component Σ O and obtain ( M, h ) ∈ ˚ F (Σ ,γ ) .We compare the original fill-in (Ω , g ) with the new fill-in ( M, h ), whichis by construction in ˚ F (Σ ,γ ) , and for which the new mean curvature H h on Σ O exceeds the original mean curvature H g pointwise; as a result,18 π Z ∂ Ω H g dσ < π Z ∂M H h dσ ≤ ˚Λ (Σ ,γ ) ,and the claim follows. (cid:3) Proof of Proposition 3.4.
Suppose Z ∂ Ω H g dσ = 8 π ˚Λ (Σ ,γ ) for some (Ω , g ) ∈ ˚ F (Σ ,γ ) . By Lemma 3.1, ∂ Ω = Σ O . We claim R ( g ) = 0,which again can be seen by applying a conformal deformation. Let u be the (unique) positive solution to(3.8) (cid:26) ∆ g u − R ( g ) u = 0 on Ω u = 1 at ∂ Ω . Consider the conformally deformed metric ˆ g = u g . Then R (ˆ g ) = 0,ˆ g = g at ∂ Ω, and H ˆ g = H g + 4 ∂u∂ν , where ν is the unit outward normalto ∂ Ω in (Ω , g ). If u is not identically a constant, then ∂u∂ν > ∂ Ωby the strong maximum principle. Hence, (Ω , ˆ g ) ∈ ˚ F (Σ ,γ ) and˚Λ (Σ ,γ ) = 18 π Z ∂ Ω H g dσ < π Z ∂ Ω H ˆ g dσ ≤ ˚Λ (Σ ,γ ) ,a contradiction. Therefore, u is constant, which shows R ( g ) = 0 on Ω.Now consider the space of metrics ¯ g on Ω given by(3.9) M g = { ¯ g | R (¯ g ) = 0 and ¯ g | T ∂ Ω = g | T ∂ Ω } , where ¯ g | T ∂ Ω denotes the induced metric on ∂ Ω from ¯ g . Since g ∈ M g maximizes the total boundary mean curvature in ˚ F , one knows that g is a critical point of the functional¯ g Z ∂ Ω H ¯ g dσ, ¯ g ∈ M g . Therefore, by [12, Corollary 2.1] (also cf. [4, Lemma 4]), g is Ricci flat,and hence is flat as Ω is 3-dimensional.It remains for us to check the topological conclusion. First we claimthat there are no closed embedded minimal surfaces (oriented or not)in the interior of Ω. We proceed by contradiction.If there were such a minimal surface T then by compactness therewould exist an interior smooth geodesic Γ : [0 , ℓ ] → Ω joining a pairof closest points between T and ∂ Ω; Γ(0) ∈ T , Γ ′ (0) ⊥ T , Γ( ℓ ) ∈ ∂ Ω,and Γ ′ ( ℓ ) ⊥ ∂ Ω. The second variation of the length of this geodesicsummed among a basis of two unit normal variations V i , i = 1 ,
2, is X i =1 δ Γ( V, V ) = − Z ℓ Ric g (Γ ′ ( s ) , Γ ′ ( s )) ds − H ∂ Ω ,g (Γ( ℓ )) − H T,g (Γ(0))= − H ∂ Ω ,g (Γ( ℓ )) < g is flat and T is minimal. This means Γ is unstable, in contra-diction with its minimizing nature. The claim follows.Theorem 1 and Proposition 1 in [10] tell us that in the absence ofinterior minimal surfaces, Ω is necessarily a handlebody with mean-convex boundary.Finally, when genus(Σ) = 0 then we know Ω a genus-0 handlebody,i.e., a 3-ball. We’ve shown its metric g is flat, so we can locally (andtherefore globally since it is simply connected) immerse Ω in R . (cid:3) F and Λ for multiple prescribed surfaces In this section we prove the theorems pertinent to the Λ functionalon compact, mean-convex, 3-manifolds with nonnegative scalar cur-vature and fixed boundary geometry consisting of possibly multiplecomponents.
Proof of Theorem 1.2 (additivity).
Let (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) , with k ≥
2. For each j = 1 , . . . , k , denote the boundary component in ∂ Ωcorresponding to Σ j by S j .Let φ < w be the unique solutionto(4.1) (cid:26) ∆ g w − R ( g ) w = φ on Ω w = 0 at ∂ Ω . For small τ >
0, consider the metrics g ( τ ) = (1 + τ w ) g . Then g ( τ ) hasstrictly positive scalar curvature on Ω, ∂ Ω has positive mean curvature otal mean curvature, scalar curvature 15 H g ( τ ) in (Ω , g ( τ ) ), and Z S j H g ( τ ) ,j dσ j ≥ Z S j H g,j dσ j − ε if τ is small enough. Here ǫ > g for this g ( τ ) .Employing Lemma 2.1, cut (Ω , ˜ g ) to isolate the boundary component S j and obtain (Ω j , ˜ g j ) ∈ ˚ F (Σ j ,γ j ) with R (˜ g j ) > H ˜ g j = H ˜ g,j on S j and ∂ Ω j = S j ∪ T j , with T j a nonempty union of smooth, stable, orientedminimal surfaces. Moreover it follows from [5] and R (˜ g j ) > T j consists of 2-spheres. Next, employ Lemma 2.2 (filling) once for everysphere in T j to replace (Ω j , ˜ g j ) with ( M j , h j ) ∈ F (Σ j ,γ j ) , with the meancurvature H h j of S j in ( M j , h j ) satisfying Z S j H h j dσ j ≥ Z S j H ˜ g j dσ j − ε ≥ Z S j H g,j dσ j − ε .Since ( M j , h j ) ∈ F (Σ j ,γ j ) , the left hand side is bounded from above byΛ (Σ j ,γ j ) . Rearranging, we have Z S j H g,j dσ j ≤ ε + 8 π Λ (Σ j ,γ j ) .Letting ε ↓
0, we obtain (1.5).Notice that if we had not let ε ↓
0, and instead carried out theprocedure above for all j = 1 , . . . , k , and summed over j , then Z ∂ Ω H g dσ ≤ kε + 8 π k X j =1 Λ (Σ j ,γ j ) .Letting ε ↓
0, and recalling that (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) was arbitrary,we conclude Λ (Σ ,γ ) ,..., (Σ k ,γ k ) ≤ k X j =1 Λ (Σ j ,γ j ) .It remains to check the reverse direction “ ≥ ” in (1.6). We will assumethat all quantities on the right are finite, as a similar argument carriesthrough to the general case. Let ε > j = 1 , . . . , k ,let (Ω j , g j ) ∈ F (Σ j ,γ j ) be such that Z ∂ Ω j H g j dσ j ≥ Λ (Σ j ,γ j ) − ε . On each Ω j , let φ j < w j be the uniquesolution to(4.2) (cid:26) ∆ g j w j − R ( g j ) w j = φ j on Ω j w j = 0 at ∂ Ω j . For small τ >
0, consider the metrics g ( τ ) j = (1 + τ w j ) g j . Then g ( τ ) j has strictly positive scalar curvature on Ω j , ∂ Ω j has positive meancurvature H g ( τ ) j in (Ω j , g ( τ ) j ), and Z ∂ Ω j H g ( τ ) j dσ j ≥ Λ (Σ j ,γ j ) − ε if τ is small enough. Applying the connect-sum construction for pos-itive scalar curvature manifolds (cf. [15] and [6]), we obtain Ω =Ω . . . k endowed with a metric g of positive scalar curvature thatcoincides with g ( τ ) j near each ∂ Ω j . In particular, (Ω , g ) ∈ F (Σ ,γ ) ,..., (Σ k ,γ k ) and it satisfies Z ∂ Ω H g dσ ≥ k X j =1 Λ (Σ j ,γ j ) − kε .Letting ε ↓
0, we conclude that “ ≥ ” holds. (cid:3) Using the tools developed so far, we can easily complete the proofsof Theorems 1.3 (finiteness) and 1.4 (rigidity).
Proof of Theorem 1.3.
Since F (Σ ,γ ) ⊂ ˚ F (Σ ,γ ) , we see that(4.3) Λ (Σ ,γ ) ≤ ˚Λ (Σ ,γ ) < ∞ for all Riemannian metrics γ on a 2-sphere Σ, where the rightmostinequality follows from Proposition 3.1. In Section 5 we will show thatthe leftmost inequality is, in fact, an equality.When k ≥
2, and γ , . . . , γ k are all metrics on a 2-sphere Σ, thenΛ (Σ ,γ ) ,..., (Σ ,γ k ) = k X j =1 Λ (Σ ,γ j ) < ∞ by Theorem 1.2 and (4.3). (cid:3) Proof of Theorem 1.4. If k = 1, the proof of Proposition 3.4 carriesthrough verbatim in this case (except we don’t need to invoke Lemma3.1, since we have no minimal boundary components).If k ≥
2, then our assumption is that18 π Z S j H g,j dσ j = Λ (Σ j ,γ j ) otal mean curvature, scalar curvature 17 holds for some fixed j ∈ { , . . . , k } , where S j represents the boundarycomponent corresponding to the Σ j on which we have equality. EmployLemma 2.1 to isolate the boundary component S j and obtain (Ω j , g j ) ∈ ˚ F (Σ j ,γ j ) , with ∂ Ω \ S j = ∅ . Then employ Lemma 2.3 to double Ω j across ∂ Ω j \ S j and obtain ( D j , h j ). Writing S ′ j for the mirror imageof S j under the doubling, we have ∂D j = S j ∪ S ′ j . By construction,( D j , h j ) ∈ F (Σ j ,γ j ) , (Σ j ,γ ′ j ) for some metric γ ′ j , so by (1.5) in Theorem 1.2,18 π Z S j H h j dσ j ≤ Λ (Σ j ,γ j ) .Also by construction, the mean curvature H h j of S j in ( D j , h j ) exceedsthe original mean curvature H g,j , so,Λ (Σ j ,γ j ) = 18 π Z S j H g,j dσ j < π Z S j H h j dσ j ≤ Λ (Σ j ,γ j ) ,a contradiction. (cid:3) Application to m (Σ; Ω)We recall and prove Theorem 1.5. Theorem.
Given a compact Riemannian 3-manifold (Ω , g ) with non-negative scalar curvature, and a mean-convex boundary Σ which is atopological 2-sphere, define m (Σ; Ω) = ˚Λ (Σ ,γ ) − π Z Σ H g dσ. Then m (Σ; Ω) is (a) well-defined and finite, (b) nonnegative, i.e., m (Σ; Ω) ≥ , and (c) m (Σ; Ω) = 0 only if (Ω , g ) is a flat 3-ball immersed in R .Finally, when Σ has positive Gauss curvature, m (Σ; Ω) = m BY (Σ; Ω) .Proof. Finiteness follows from Proposition 3.1, and nonnegativity istrue by the definition of ˚Λ (Σ ,γ ) . The rigidity statement follows fromProposition 3.4. Finally, when Σ has positive Gauss curvature, Theo-rem 1.1 implies Z Σ H dσ = sup ( M,h ) ∈F (Σ ,γ ) Z ∂M H h dσ = 8 π Λ (Σ ,γ ) and the result follows by Proposition 5.1 below. (cid:3) Proposition 5.1.
Let Σ be a closed, connected, orientable surface witha Riemannian metric γ . Then Λ (Σ ,γ ) = ˚Λ (Σ ,γ ) .In other words, ˚ F (Σ ,γ ) and ˚Λ (Σ ,γ ) can be replaced by F (Σ ,γ ) and Λ (Σ ,γ ) in the definition of m (Σ; Ω) .Proof. It suffices to show ˚Λ (Σ ,γ ) ≤ Λ (Σ ,γ ) . Pick (Ω , g ) ∈ ˚ F (Σ ,γ ) such thatΣ H = ∂ Ω \ Σ O is nonempty. Let h be a metric given by Lemma 2.3on the doubling D of Ω across Σ H . Then H h > H g on Σ O . ApplyingTheorem 1.2 to ( D, h ) at Σ O , we have Z Σ O H h dσ ≤ π Λ (Σ ,γ ) . Therefore, Z ∂ Ω H g dσ = Z Σ O H g dσ < Z Σ O H h dσ ≤ π Λ (Σ ,γ ) , which proves the claim. (cid:3) References [1] J. D. Brown, J. W. York, Jr.,
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E-mail address : [email protected] (Pengzi Miao) Department of Mathematics, University of Miami, CoralGables, FL 33146, USA.
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