We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.
Accepted:
Online First:
Published online:
DOI: 10.1007/s10240-012-0047-5
Simon Brendle 1
@article{PMIHES_2013__117__247_0,
author = {Simon Brendle},
title = {Constant mean curvature surfaces in warped product manifolds},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {247--269},
year = {2013},
publisher = {Springer-Verlag},
volume = {117},
doi = {10.1007/s10240-012-0047-5},
mrnumber = {3090261},
zbl = {1273.53052},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-012-0047-5/}
}
TY - JOUR AU - Simon Brendle TI - Constant mean curvature surfaces in warped product manifolds JO - Publications Mathématiques de l'IHÉS PY - 2013 SP - 247 EP - 269 VL - 117 PB - Springer-Verlag UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-012-0047-5/ DO - 10.1007/s10240-012-0047-5 LA - en ID - PMIHES_2013__117__247_0 ER -
%0 Journal Article %A Simon Brendle %T Constant mean curvature surfaces in warped product manifolds %J Publications Mathématiques de l'IHÉS %D 2013 %P 247-269 %V 117 %I Springer-Verlag %U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-012-0047-5/ %R 10.1007/s10240-012-0047-5 %G en %F PMIHES_2013__117__247_0
Simon Brendle. Constant mean curvature surfaces in warped product manifolds. Publications Mathématiques de l'IHÉS, Volume 117 (2013), pp. 247-269. doi: 10.1007/s10240-012-0047-5
[1.] Uniqueness theorems for surfaces in the large I, Vestn. Leningr. Univ., Volume 11 (1956), pp. 5-17 | MR
[2.] Stability of hypersurfaces with constant mean curvature, Math. Z., Volume 185 (1984), pp. 339-353 | MR | Zbl | DOI
[3.] Energy in general relativity, Tsing Hua Lectures on Geometry and Analysis (Hsinchu 1990–1991), International Press, Cambridge (1997), pp. 5-27 | MR | Zbl
[4.] Einstein Manifolds, Classics in Mathematics, Springer, Berlin, 2008 | Zbl | MR
[5.] H. Bray, The Penrose Inequality in General Relativity and Volume Comparison Theorems Involving Scalar Curvature, Ph.D. Thesis, Stanford University, 1997. | MR
[6.] An isoperimetric comparison theorem for Schwarzschild space and other manifolds, Proc. Am. Math. Soc., Volume 130 (2002), pp. 1467-1472 | MR | Zbl | DOI
[7.] S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differ. Geom., to appear. | MR | Zbl
[8.] Some remarks on the quasi-local mass, Mathematics and General Relativity (Contemporary Mathematics, 71), Am. Math. Soc., Providence (1986), pp. 9-14 | MR | Zbl | DOI
[9.] Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys., Volume 214 (2000), pp. 137-189 | MR | Zbl | DOI
[10.] On isoperimetric surfaces in general relativity, Pac. J. Math., Volume 231 (2007), pp. 63-84 | MR | Zbl | DOI
[11.] M. Eichmair and J. Metzger, Large isoperimetric surfaces in initial data sets, J. Differ. Geom., to appear. | MR | Zbl
[12.] On large volume preserving stable CMC surfaces in initial data sets, J. Differ. Geom., Volume 91 (2012), pp. 81-102 | MR | Zbl
[13.] M. Eichmair and J. Metzger, Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions, . | arXiv | MR | Zbl
[14.] On the subharmonicity and plurisubharmonicity of geodesically convex functions, Indiana Univ. Math. J., Volume 22 (1973), pp. 641-653 | MR | Zbl | DOI
[15.] C ∞ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. Éc. Norm. Super., Volume 12 (1979), pp. 47-84 | MR | Zbl | Numdam
[16.] A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Éc. Norm. Super., Volume 11 (1978), pp. 451-470 | MR | Zbl | Numdam
[17.] Dirac operators on hypersurfaces of manifolds with negative scalar curvature, Ann. Glob. Anal. Geom., Volume 23 (2003), pp. 247-264 | MR | Zbl | DOI
[18.] The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., Volume 59 (2001), pp. 353-437 | MR | Zbl
[19.] Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math., Volume 124 (1996), pp. 281-311 | MR | Zbl | DOI
[20.] Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J., Volume 48 (1999), pp. 711-748 | MR | Zbl | DOI
[21.] Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures, Differential Geometry (Pitman Monographs and Surveys in Pure and Applied Mathematics, 52), Longman, Harlow (1991), pp. 279-296 | MR | Zbl
[22.] Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds, Geom. Funct. Anal., Volume 19 (2009), pp. 910-942 | MR | Zbl | DOI
[23.] Constant mean curvature spheres in Riemannian manifolds, Manuscr. Math., Volume 128 (2008), pp. 275-295 | MR | Zbl | DOI
[24.] Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, New York, 2006 | Zbl | MR
[25.] On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds, J. Am. Math. Soc., Volume 20 (2007), pp. 1091-1110 | MR | Zbl | DOI
[26.] M. Reiris, Static solutions from the point of view of comparison geometry, preprint (2011). | MR | Zbl
[27.] The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature, Manuscr. Math., Volume 113 (2004), pp. 403-421 | MR | Zbl | DOI
Cited by Sources: