Rigidity of marginally outer trapped (hyper)surfaces with negative 𝜎-constant

Mendes, Abraão

doi:10.1090/tran/7752

Rigidity of marginally outer trapped (hyper)surfaces with negative $\sigma$-constant
HTML articles powered by AMS MathViewer

by Abraão Mendes PDF

Trans. Amer. Math. Soc. 372 (2019), 5851-5868 Request permission

Abstract:

In this paper we generalize a result of Galloway and Mendes in two different situations: in the first case for marginally outer trapped surfaces (MOTSs) of genus greater than $1$ and, in the second case, for MOTSs of high dimension with negative $\sigma$-constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension $2$) or for the volume (in dimension $\ge 3$). These results are extensions of theorems of Nunes and Moraru to general (non-time-symmetric) initial data sets.

References

Lucas C. Ambrozio, Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds, J. Geom. Anal. 25 (2015), no. 2, 1001–1017. MR 3319958, DOI 10.1007/s12220-013-9453-2
L. Andersson, M. Mars, and W. Simon, Local existence of dynamical and trapping horizons, Phys. Rev. Lett. 95 (2005), 111102.
Lars Andersson, Marc Mars, and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys. 12 (2008), no. 4, 853–888. MR 2420905, DOI 10.4310/ATMP.2008.v12.n4.a5
Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
Hubert Bray, Simon Brendle, and Andre Neves, Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830. MR 2765731, DOI 10.4310/CAG.2010.v18.n4.a6
Mingliang Cai, Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999) Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1–7. MR 1925731, DOI 10.2969/aspm/03410001
Mingliang Cai and Gregory J. Galloway, Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature, Comm. Anal. Geom. 8 (2000), no. 3, 565–573. MR 1775139, DOI 10.4310/CAG.2000.v8.n3.a6
Mingliang Cai and Gregory J. Galloway, On the topology and area of higher-dimensional black holes, Classical Quantum Gravity 18 (2001), no. 14, 2707–2718. MR 1846368, DOI 10.1088/0264-9381/18/14/308
Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. MR 562550, DOI 10.1002/cpa.3160330206
Gregory J. Galloway, Rigidity of marginally trapped surfaces and the topology of black holes, Comm. Anal. Geom. 16 (2008), no. 1, 217–229. MR 2411473, DOI 10.4310/CAG.2008.v16.n1.a7
Gregory J. Galloway, Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 221–239. MR 2906927
Gregory J. Galloway and Abraão Mendes, Rigidity of marginally outer trapped 2-spheres, Comm. Anal. Geom. 26 (2018), no. 1, 63–83. MR 3761653, DOI 10.4310/CAG.2018.v26.n1.a2
G. J. Galloway and N. Ó. Murchadha, Some remarks on the size of bodies and black holes, Classical Quantum Gravity 25 (2008), no. 10, 105009.
Gregory J. Galloway and Richard Schoen, A generalization of Hawking’s black hole topology theorem to higher dimensions, Comm. Math. Phys. 266 (2006), no. 2, 571–576. MR 2238889, DOI 10.1007/s00220-006-0019-z
G. W. Gibbons, Some comments on gravitational entropy and the inverse mean curvature flow, Classical Quantum Gravity 16 (1999), no. 6, 1677–1687. MR 1697098, DOI 10.1088/0264-9381/16/6/302
Jerry L. Kazdan and F. W. Warner, Prescribing curvatures, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 309–319. MR 0394505
Osamu Kobayashi, Scalar curvature of a metric with unit volume, Math. Ann. 279 (1987), no. 2, 253–265. MR 919505, DOI 10.1007/BF01461722
Mario Micallef and Vlad Moraru, Splitting of 3-manifolds and rigidity of area-minimising surfaces, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2865–2872. MR 3336611, DOI 10.1090/S0002-9939-2015-12137-5
Vlad Moraru, On area comparison and rigidity involving the scalar curvature, J. Geom. Anal. 26 (2016), no. 1, 294–312. MR 3441515, DOI 10.1007/s12220-014-9550-x
Ivaldo Nunes, Rigidity of area-minimizing hyperbolic surfaces in three-manifolds, J. Geom. Anal. 23 (2013), no. 3, 1290–1302. MR 3078354, DOI 10.1007/s12220-011-9287-8
Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
E. Woolgar, Bounded area theorems for higher-genus black holes, Classical Quantum Gravity 16 (1999), no. 9, 3005–3012. MR 1713392, DOI 10.1088/0264-9381/16/9/316
Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546