RIGIDITY OF TIME-FLAT SURFACES IN THE MINKOWSKI SPACETIME

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1 RIGIDITY OF TIME-FLAT SURFACES IN THE MINKOWSKI SPACETIME PO-NING CHEN, MU-TAO WANG, AND YE-KAI WANG Abstract. A time-flat condition on spacelike 2-surfaces in spacetime is considered ere. Tis condition is analogous to constant torsion condition for curves in tree dimensional space and as been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In tis article, we address te question in te title and prove several local and global rigidity teorems for suc surfaces in te Minkowski spacetime. 1. Introduction Te geometry of spacelike 2-surfaces in spacetime plays a crucial role in general relativity. Penrose s singularity teorem predicts future singularity formation from te existence of a trapped surface. A black ole is quasi-locally described by a marginally outer trapped surface. Tese conditions can be expressed in terms of te mean curvature vector field H of te 2-surface. H is te unique normal vector field determined by te variation of te area functional and is ultimately connected to te warping of spacetime in te vicinity of te 2-surface. It is tus not surprising tat several definitions of quasi-local mass in general relativity are closely related to te mean curvature vector field. In particular, bot te Hawking mass [7] and te Brown-York-Liu-Yau mass [3, 8] involve te norm of te mean curvature vector field H. In te new definition of quasi-local mass in [12, 13], in addition to H, te direction of te mean curvature vector field is also utilized. Wen te mean curvature vector field is spacelike everywere on tus H > 0), H defines a connection one-form α H of te normal bundle see Definition 2 for te precise definition of α H ). Te quasi-local mass in [12, 13] is defined in terms of te induced metric σ on te surface, H, and α H. In particular, te condition 1.1) div σ α H ) = 0 implies tat te isometric embedding of into R 3 R 3,1 is an optimal embedding in te sense of [12, 13]. Recently, Bray and Jauregui [2] discovered a very interesting monotonicity property of te Hawking mass along surfaces tat satisfy te condition 1.1). Suc surfaces are said to be time-flat in [2] and include all 2-surfaces in a time-symmetric initial data Date: October 24, P.-N. Cen is supported by NSF grant DMS and M.-T. Wang is supported by NSF grant DMS All autors would like to tank Hug Bray for raising te question in te title and for inspiring discussions during is visit to Columbia Mat Department on October 11,

2 2 PO-NING CHEN, MU-TAO WANG, AND YE-KAI WANG set. A natural rigidity question raised by Bray [1] is Must suc a surface in te Minkowski spacetime lie in a totally geodesic R 3? Due to te existence of non-flat curves in R 3 wit constant torsion, some global condition needs to be imposed in order for te rigidity property to old. In tis article, we prove several global and local rigidity teorems for time-flat surfaces in te Minkowski spacetime under various conditions. Te local rigidity teorem olds in any iger dimensional Minkowski spacetime: Teorem 4. Suppose is a mean convex ypersurface wic lies in a totally geodesic R n in te n+1 dimensional Minkowski spacetime R n,1, ten is locally rigid as a timeflat n 1 dimensional submanifold in R n,1. In oter words, any infinitesimal deformation of tat preserves te time-flat condition must be a deformation in te R n direction, a deformation tat is induced by a Lorentz transformation of R n,1, or a combination of tese two types of deformations. We also proved two global rigidity teorems: Teorem 5. Suppose is a time-flat 2-surface in R 3,1 suc tat α H = 0 and is a topological spere, ten lies in a totally geodesic yperplane. Teorem 6. Suppose is a time-flat 2-surface in R 3,1 suc tat 1) Te induced metric on is axially symmetric and of positive Gaussian curvature. 2) can be written as te grap of an axially symmetric function τ over a convex surface in R 3. Ten lies in a total geodesic yperplane in R 3,1. Teorem 4 follows from applying te Reilly formula to te linearized equation 3.7). Teorem 6 uses a mean curvature comparison lemma derived in [5] and te minimizing property of critical points of te Wang-Yau energy. Teorem 4 is proved in 3, Teorem 5 is proved in 4, and Teorem 6 is proved in Geometry of spacelike 2-surface in spacetime Let N be a time-oriented spacetime. Denote te Lorentzian metric on N by, and covariant derivative by N. Let be a closed space-like two-surface embedded in N. Denote te induced Riemannian metric on by σ and te gradient and Laplace operator of σ by and, respectively. Given any two tangent vector X and Y of, te second fundamental form of in N is given by IIX,Y) = N X Y) were ) denotes te projection onto te normal bundle of. Te mean curvature vector is te trace of te second fundamental form, or H = tr II = 2 a=1 IIe a,e a ) were {e 1,e 2 } is an ortonormal basis of te tangent bundle of. Te normal bundle is of rank two wit structure group SO1,1) and te induced metric on te normal bundleis of signature,+). Since te Lie algebra of SO1,1) is isomorpic to R, te connection form of te normal bundle is a genuine 1-form tat depends on te

3 TIMEFLAT SURFACE 3 coice of te normal frames. Te curvature of te normal bundle is ten given by an exact 2-form wic reflects te fact tat any SO1, 1) bundle is topologically trivial. Connections of different coices of normal frames differ by an exact form. We define see [13]) Definition 1. Let e 3 be a space-like unit normal along, te connection one-form determined by e 3 is defined to be 2.1) α e3 = N ) e 3,e 4 were e 4 is te future-directed time-like unit normal tat is ortogonal to e 3. Definition 2. Suppose te mean curvature vector field H of in N is a spacelike vector field. Te connection one-form in mean curvature gauge is α H = N ) e 3,e 4, were e 3 = H H and e 4 is te future-directed timelike unit normal tat is ortogonal to e Local rigidity of mean convex ypersurfaces in R n R n,1 Te local rigidity teorem olds in iger dimensional Minkowski spacetime as well. Let be a closed embedded spacelike codimension-2 submanifold in Minkowski spacetime R n,1 and σ be its induced metric. Suppose tat te mean curvature vector H of is spacelike. Let e n = H H and e n+1 be te unit future timelike normal tat is ortogonal to e n. Let α H = ) e n,e n+1 be te connection one-form on te normal bundle of determined by te mean curvature gauge. Definition 3. We say is time-flat if div σ α H ) = 0. Te local rigidity problem can be formulated as follows. Suppose is time-flat and is given by an embedding X. Suppose V is a smoot vector field along suc tat te image of Xs) = X+sV is infinitesimally time-flat in te sense te derivatives of div σ α H ) along te image wit respect to s is zero wen s = 0. Do all suc V correspond to trivial deformations? Itis easy to see tat submanifoldslying in atotally geodesic slice is time-flat. We assume Ω = {t = 0} = R n. It is clear tat any deformation in te R n direction preserves te time-flat condition. On te oter and, a Lorentz transformation preserves te geometry of and tus preserves te time-flat condition. Let, denote te covariant derivative and Laplacian of te induced metric σ. Let ab, be te second fundamental form and mean curvature of R n wit respect to te outward unit normal ν. Teorem 4. Suppose is a mean convex ypersurface wic lies in a totally geodesic R n in te n+1 dimensional Minkowski spacetime R n,1, ten is locally rigid as a timeflat n 1 dimensional submanifold in R n,1. In oter words, any infinitesimal deformation of tat preserves te time-flat condition must be a deformation in te R n direction, a deformation tat is induced by a Lorentz transformation of R n,1, or a combination of tese two types of deformations.

4 4 PO-NING CHEN, MU-TAO WANG, AND YE-KAI WANG Proof. In tis proof, we denote α H by α. Since δdiv σ α) depends linearly on infinitesimal deformation and any deformation in R n corresponds to trivial deformations, it suffices to consider deformations in te time direction. Let V = f t for a smoot function f defined on be suc an infinitesimal deformation and Xs) = τs),x 1 s),...,x n s)) be te corresponding deformation. Since we only vary te surface in te time direction, X i s) = X i 0) for i = 1,...,n, and s s=0 τs) = f. We start by computing te variation of div σ α. Te induced metrics satisfy 3.1) σs) ab = σ ab τs) τs) u a u b. Since τ0) = 0, δσ = 0. Let s be te Laplacian of te induced metric on Xs). We ave 3.2) H = s τs), s X 1,..., s X n ). δσ = 0 implies te infinitesimal variation of Laplacian is zero. Terefore, we ave 3.3) δh = f) t 3.4) δ H 2 = 2 δh, e n = 0 δe n = δh + δ H 2 H = f 3.5) t From 0 = δ e n+1, u = δ e a n+1,e n, we ave 3.6) δe n+1 = f f e n Putting tese facts togeter, we get and δα) a = δ D a e n,e n+1 = D f u a t = a f e n,e n+1 + D a f ) + ab b f 3.7) δdiv σ α) = ) t, t + D ae n, f + fe n ) f + a ) ab b f. We remark tat te linearization of tis operator was also derived in [4, 9]. Let u solve te Diriclet problem { u = 0 in Ω u = f on We recall te Reilly formula on R n 3.8) D 2 u 2 = ab a f b f +2 fe n u)+e n u)) 2). Ω

5 TIMEFLAT SURFACE 5 Tis was used in [9] to derive minimizing property of te Wang-Yau quasi-local energy. On te oter and, multiply δdiv σ α) = 0 by f and integrate over to get f) 2 3.9) ab a f b f = 0. Adding 3.8) and 3.9) togeter and completing square, we obtain f D 2 u e n u)) = 0. Hence u is a linear function up to a constant. Ω 4. Global rigidity for surfaces wit α H = 0 Te global rigidity olds true if one assumes in addition tat α H = 0. Teorem 5. Suppose is a time-flat 2-surface in R 3,1 suc tat α H = 0 and is a topological spere, ten lies in a totally geodesic R 3 Proof. Denote by e 3 = H H and e 4 to be te unit future timelike normal tat is ortogonal to e 3. Te second fundamental form of can be written as 3 ab e 3 4 ab e 4 and 4 ab is tracefree. Te Codazzi equation for 4 ab reads a 4 ab btr σ 4 +α H ) a 4 ab tr σ 4 α H ) b = 0. Since α H = 0 and 4 ab is trace-free, tis reduces to a 4 ab = 0. A divergence-free symmetric trace-free 2-tensor corresponds to a olomorpic quadratic differential, wic must vanis since is a topological spere. In particular, 3 ab and 4 ab can be diagonalized simultaneously. By [14], it follows tat lies in a umbilical ypersurface in R 3,1. If lies in a totally geodesic R 3 ten te proof is finised. If lies in a yperbola in R 3,1 ten te position vector is a unit timelike normal of and te connection form determined by te position vector is also zero. Hence, te angle between e 4 and te position vector is constant. It follows tat as constant mean curvature in te yperbola. Hence, is a round spere in te yperbola and tus te intersection of a totally geodesic R 3 wit te yperbola. 5. Wang Yau quasi-local energy. In te next section, te positivity of Wang Yau quasi-local energy is used to prove te global rigidity of time-flat axially symmetric surfaces in R 3,1. We recall te definition of Wang Yau quasi-local energy in tis section. Let be a spacelike surface in a spacetime N. Te definition of Wang Yau quasi-local energy relies on te pysical data on wic consist of te induced metric σ, te norm of te mean curvature vector H > 0, and te connection one-form α H. Given te triple of pysical data {σ, H,α H }, one assigns a quasi-local energy for eac pairof isometric embeddingx of intor 3,1 andfuture-directed

6 6 PO-NING CHEN, MU-TAO WANG, AND YE-KAI WANG unit timelike vector T 0 in R 3,1. In terms of τ = X T 0, te quasi-local energy is defined to be [ 5.1) E,τ) = Ĥdv 1+ τ 2 cosθ H τ θ α H τ)] dv, were and aretecovariant derivative andlaplaceoperator wit respectto σ, andθ is τ defined by sinθ = H 1+ τ 2. Finally, is te projection of X) onto te complement of T 0 and Ĥ is te mean curvature of in R 3. In E,τ), te first argument represents a pysical surface in spacetime wit te data σ, H,α H ), wile te second argument τ indicates an isometric embedding of te induced metric into R 3,1 wit time function τ wit respect to T 0. As remarked in [5], no information is lost by coosing a fixed future-directed unit timelike vector T 0 and considering E,τ) as a energy functional on functions on. 6. Global rigidity for axially symmetric and time-flat surfaces in R 3,1. In tis section, we prove te following teorem for te global rigidity of time-flat axially symmetric surfaces in R 3,1. Teorem 6. Suppose is a time-flat 2-surface in R 3,1 suc tat 1) Te induced metric on is axially symmetric and of positive Gaussian curvature. 2) can be written as te grap of an axially symmetric function τ over a convex surface in R 3. Ten lies in a total geodesic yperplane in R 3,1. Proof. Let σ be te induced metric on and K be te Gaussian curvature of σ. By condition 1) and te isometric embedding teorem of Nirenberg and Pogorelov [10, 11], tere exists a unique isometric embedding of into R 3. Denote te image by 0, wic is a convex surface in R 3. From condition 2) and equation 3.3) of [13], it follows K + det 2 τ) 1+ τ 2 > 0. Tis togeter wit K > 0 implies tat, for any 0 s 1, K + det 2 sτ) 1+ sτ 2 > 0. Hence, tere is a unique isometric embedding of te surface into R 3,1 wit time function sτ for any 0 s 1 by Teorem 3.1 of [13]. Denote te image by s and its mean curvature vector by H s. Up to a Lorentz transformation, we can assume tat 1 =. Let E 0,sτ) be te quasi-local energy for te surface 0 wit reference embedding s. Lemma 3 of [5] gives te following inequality about te mean curvature vectors of s 6.1) H 0 H s for any 0 s 1. Namely, for axially symmetric surfaces wit te same induced metric in R 3,1, te isometric embedding into R 3 as te largest norm of te mean curvature vector.

7 TIMEFLAT SURFACE 7 We prove te teorem by combining equation 6.1) wit an inequality obtained from te positivity of Wang Yau quasi-local energy. We treat 0 in te totally geodesic R 3 in R 3,1 as a surface in a spacetime. Namely, it comes wit te pysical data {σ, H 0,0}. We consider te Wang Yau quasi-local energy for oter isometric embedding of 0 into R 3,1. Fixing a future-directed unit timelike vector T 0 in R 3,1, te quasi-local energy becomes an functional E 0,f) for functions on 0. Since te data {σ, H 0,0} comes from te surface 0 embedded in R 3,1, E 0,0) = 0 and it is natural to expect tat quasi-local energy for oter isometric embedding is non-negative. Namely, one expects tat f = 0 is te global minimum of te quasi-local energy functional E 0,f). Te following lemma sows tat tis olds in axially symmetric cases. Lemma 7. Let 0 be any convex and axially symmetric surface in a totally geodesic R 3 in R 3,1 and K be te Gaussian curvature of te induced metric σ. Suppose τ is an axially symmetric function on 0 suc tat K + det 2 τ) 1+ τ 2 > 0. Ten E 0,τ) 0. Namely, f = 0 is te global minimum of te quasi-local energy functional E 0,f) among all axially symmetric time functions. Proof. Tis follows from Lemma 2 of [5] and equation 6.1). By Proposition 3.2 of [12], E 0,τ) can be written as E 0,τ) = 1+ τ 2 ) H τ) 2 τ sin 1 τ H 1 1+ τ 2) α H 1 τ) 1+ τ 2 ) H τ) 2 τ sin 1 τ H 0 1+ τ 2) α H 0 τ). Since is time-flat and α H0 = 0, it follows tat E 0,τ) = 1+ τ 2 ) H τ) 2 τ sin 1 τ H 1 1+ τ 2) 6.2) 1+ τ 2 ) H τ) 2 τ sin 1 τ H 0 1+ τ 2) 0 Combining equation 6.1) and 6.2), it follows tat H 0 = H 1 and E 0,τ) = 0. From te proof of Lemma 2 of [5], it follows tat E 0,sτ) = 0 for all s. In particular, s=0 E 0,sτ) = 0 2 s Applying Teorem 1.3 and Lemma 2.1 of [9] to te critical point f = 0 of te quasi-local energy functional E 0,f), one concludes tat τ is a linear combination of te coordinate functions of 0 up to a constant. It follows tat lies in a totally geodesic R 3 in R 3,1.

8 8 PO-NING CHEN, MU-TAO WANG, AND YE-KAI WANG A corollary of Teorem 6 is a uniqueness teorem for isometric embeddings of an axially symmetric metric into R 3,1. Corollary 8. Let σ be an axially-symmetric metric on S 2 wit positive Gaussian curvature. Suppose X is a time-flat isometric embedding of σ into R 3,1 suc tat te image can be written as te grap of an axially-symmetric function over a convex surface in R 3. Ten X must lie in a totally geodesic yperplane. In view of te classical rigidity teorem of isometric embeddings[6] for metrics of positive Gaussian curvature into R 3, te following conjecture is a natural extension by imposing te additional time-flat condition to make te problem a well-determined system: Conjecture 9. Let σ be a metric on S 2 wit positive Gaussian curvature. Suppose X is a time-flat isometric embedding of σ into R 3,1 suc tat te image can be written as te grap over a convex surface in R 3. Ten te image of X must lie in a totally geodesic yperplane. References [1] H. Bray, Personal communications. [2] H. Bray and J. Jauregui, preprint. [3] J. D. Brown and J. W. York, Quasilocal energy in general relativity, Matematical aspects of classical field teory Seattle, WA, 1991), , Contemp. Mat. 132, Amer. Mat. Soc., Providence, RI, [4] P. Cen, M.-T. Wang, and S.-T. Yau, Evaluating quasilocal energy and solving optimal embedding equation at null infinity, Comm. Mat. Pys ), no.3, [5] P. Cen, M.-T. Wang,and S.-T. Yau, Minimizing properties of critical points of quasi-local energy, to appear in Comm. Mat. Pys. arxiv: [6] S. Con-Vossen, Unstarre gesclossene Fläcen, German) Mat. Ann ), no. 1, [7] S. W. Hawking, Gravitational radiation in an expanding universe, J. Mat. Pys., 9, , 1968). [8] C.C. Liu and S.T. Yau, Positivity of quasilocal mass, Pys. Rev. Lett. 90, ). [9] P. Miao, L.-F. Tam and N.Q. Xie, Critical points of Wang-Yau quasi-local energy. Ann. Henri Poincaré. 12 5): , 2011 [10] L. Nirenberg, Te Weyl and Minkowski problems in differential geometry in te large, Comm. Pure Appl. Mat. 6, 1953), [11] A. V. Pogorelov, Regularity of a convex surface wit given Gaussian curvature, Russian) Mat. Sbornik N.S. 3173), 1952) [12] M.-T. Wang and S.-T. Yau, Quasilocal mass in general relativity, Pys. Rev. Lett ), no. 2, no [13] M.-T. Wang and S.-T. Yau, Isometric embeddings into te Minkowski space and new quasi-local mass, Comm. Mat. Pys ), no. 3, [14] S. T. Yau, Submanifolds wit constant mean curvature. I, II, Amer. J. Mat ), ; ibid ),