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Velásquez, Marco Antonio Lázaro

The geometry at infinity of LW-spacelike submanifolds in semi-Riemannian space forms. (English) Zbl 1551.53067

Colloq. Math. 175, No. 2, 245-261 (2024).
Let \(\mathbb{Q}^{n+p}_{p}(c)\) be the \((n + p)\)-dimensional connected with index \(p\) and constant sectional curvature \(c\in \mathbb{R}\). An \(n\)-dimensional submanifold \(M^{n}\) immersed in \(\mathbb{Q}^{n+p}_{p}(c)\) is said to be space-like if the on \(M^{n}\) is positive definite. A space-like submanifold of \(\mathbb{Q}^{n+p}_{p}(c)\) is said to be linear Weingarten (for short, an LW-space-like submanifold) when its mean and normalized scalar curvatures are linearly related.
The paper under review studies complete noncompact LW-space-like submanifolds in \(\mathbb{Q}^{n+p}_{p}(c)\) with parallel normalized mean curvature vector. Under suitable restrictions on the behavior of the mean curvature at infinity and values of the norm of the traceless part of the , the authors show that these surfaces must be isometric to one of the following :
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\(\mathbb{R}^{n-1} \times \mathbb{H}^{1}(c_2)\), with \(c_2 < 0\), when \(c = 0\);
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\(\mathbb{S}^{n-1}(c_1) \times \mathbb{H}^{1}(c_2)\), with \(c_1 > 0\), \(c_2 < 0\) and \(1/c_1 + 1/c_2 = 1/c\), when \(c > 0\);
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\(\mathbb{H}^{n-1}(c_1) \times \mathbb{H}^{1}(c_2)\), with \(c_1 < 0\), \(c_2 < 0\) and \(1/c_1 + 1/c_2 = 1/c\), when \(c < 0\).
Reviewer: Andrea Tamburelli (Houston)

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

semi-Riemannian space forms; space-like submanifolds; linear Weingarten submanifolds; parallel normalized mean curvature vector; hyperbolic cylinders
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[1] N. Abe, N. Koike and S. Yamaguchi, Congruence theorems for proper semi-Riemannian hypersurfaces in a real space form, Yokohama Math. J. 35 (1987), 123-136. · Zbl 0645.53010
[2] K. Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), 13-19. · Zbl 0611.53047
[3] L. J. Alías, A. Caminha and F. Y. do Nascimento, A maximum principle at infinity with applications to geometric vector fields, J. Math. Anal. Appl. 474 (2019), 242-247. · Zbl 1414.53028
[4] L. J. Alías and A. Romero, Integral formulas for compact spacelike n-submani-folds in de Sitter spaces. Applications to the parallel mean curvature vector case, Manuscripta Math. 87 (1995), 405-416. · Zbl 0838.53043
[5] F. E. C. Camargo, R. M. B. Chaves and L. A. M. Sousa Jr., New characteriza-tions of complete spacelike submanifolds in semi-Riemannian space forms, Kodai Math. J. 32 (2009), 209-230. · Zbl 1171.53032
[6] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Math. Proc. Cambridge Philos. Soc. 82 (1977), 489-495.
[7] L. Cao and G. Wei, A new characterization of hyperbolic cylinder in anti-de Sitter space H n+1 1 (-1), J. Math. Anal. Appl. 329 (2007), 408-414. · Zbl 1112.53052
[8] X. Chao, On complete spacelike submanifolds in semi-Riemannian space forms with parallel normalized mean curvature vector, Kodai Math. J. 34 (2011), 42-54. · Zbl 1221.53075
[9] R. M. B. Chaves, L. A. M. Sousa Jr. and B. C. Valério, New characterizations for hyperbolic cylinders in anti-de Sitter spaces, J. Math. Anal. Appl. 393 (2012), 166-176. · Zbl 1243.53016
[10] Q. M. Cheng, Complete space-like submanifolds with parallel mean curvature vector, Math. Z. 206 (1991), 333-339. · Zbl 0695.53042
[11] S. Y. Cheng and S. T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski space, Ann. of Math. 104 (1976), 407-419. · Zbl 0352.53021
[12] S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195-204. · Zbl 0349.53041
[13] M. Dajczer and K. Nomizu, On flat surfaces in S 3 1 and H 3 1 , in: Manifolds and Lie Groups (Notre Dame, IN, 1980), Birkäuser, Boston, 1981, 71-108. · Zbl 0485.53047
[14] H. F. de Lima, F. R. dos Santos and M. A. L. Velásquez, New characterizations of hyperbolic cylinders in semi-Riemannian space forms, J. Math. Anal. Appl. 434 (2016), 765-779. · Zbl 1329.53036
[15] G. J. Galloway and J. M. M. Senovilla, Singularity theorems based on trapped submanifolds of arbitrary co-dimension, Class. Quantum Grav. 27 (2010), art. 152002, 10 pp. · Zbl 1195.83065
[16] A. J. Goddard, Some remarks on the existence of spacelike hypersurfaces of constant mean curvature, Math. Proc. Cambridge Philos. Soc. 82 (1977), 489-495. · Zbl 0386.53042
[17] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime, Cambridge Univ. Press, London, 1973. · Zbl 0265.53054
[18] T. Ishihara, Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature, Michigan Math. J. 35 (1988), 345-352. · Zbl 0682.53055
[19] H. Li, Complete spacelike submanifolds in de Sitter space with parallel mean curvature vector satisfying H 2 = 4(n -1)/n 2 , Ann. Global Anal. Geom. 15 (1997), 335-345. · Zbl 0889.53043
[20] H. Li, Y. J. Suh and G. Wei, Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc. 46 (2009), 321-329. · Zbl 1165.53361
[21] Z. Liang and X. Zhang, Spacelike hypersurfaces with negative total energy in de Sitter spacetime, J. Math. Phys. 53 (2012), art. 022502, 10 pp. · Zbl 1274.83028
[22] J. Marsden and F. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep. 66 (1990), 109-139.
[23] S. Montiel, An integral inequality for compact spacelike hypersurfaces in the de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988), 909-917. · Zbl 0677.53067
[24] S. Montiel, A characterization of hyperbolic cylinders in the de Sitter space, Tôhoku Math. J. 48 (1996), 23-31. · Zbl 0848.53039
[25] S. Nishikawa, On spacelike hypersurfaces in a Lorentzian manifold, Nagoya Math. J. 95 (1984), 117-124. · Zbl 0544.53050
[26] R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965), 57-59. · Zbl 0125.21206
[27] O. Perdomo, New examples of maximal space like surfaces in the anti-de Sitter space, J. Math. Anal. Appl. 353 (2009), 403-409. · Zbl 1163.83008
[28] J. Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ. Math. J. 36 (1987), 349-359. · Zbl 0626.53041
[29] W. Santos, Submanifolds with parallel mean curvature vector in sphere, Tôhoku Math. J. 46 (1994), 403-415. · Zbl 0812.53053
[30] J. M. M. Senovilla, Singularity theorems in General Relativity: achievements and open questions, in: Einstein and the Changing Worldviews of Physics, Birkhäuser Boston, Boston, MA, 2012, 305-316. · Zbl 1229.83018
[31] S. Stumbles, Hypersurfaces of constant mean extrinsic curvature, Ann. Phys. 133 (1980), 28-56. · Zbl 0472.53063
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