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Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity. (English) Zbl 07276077
Summary: Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type \(M^n\times_{\rho}\mathbb{R}_1\) whose Riemannian base \(M^n\) has nonnegative sectional curvature and the warping function \(\rho\) is convex on \(M^n\), we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in \(M^2\times_{\rho}\mathbb{R}_1\) and we obtain uniqueness results for entire Killing graphs constructed over \(M^n\).
MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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References:
[1] [1]Ahlfors L. V.,Sur le type dune surface de Riemann. C. R. Acad. Sc. Paris 201(1935), 30-32. · JFM 61.0365.01
[2] [2]Albujer A. L.,New examples of entire maximal graphs inH2×R1. Diff. Geom. Appl.26(2008), 456-462. · Zbl 1147.53047
[3] [3]Albujer A. L., Aledo J. A. and Al´ıas L. J.,On the scalar curvature of in hypersurfaces in spaces with Killing field. Adv. Geom.10(2010), 487-503. · Zbl 1194.53018
[4] [4]Albujer A. L. and Al´ıas L. J.,Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces. J. Geom. Phys.59(2009), 620-631. · Zbl 1173.53025
[5] [5]Albujer A. L. and Al´ıas L. J.,Parabolicity of maximal surfaces in Lorentzian product spaces. Math. Z.267(2011), 453-464. · Zbl 1218.53062
[6] [6]Albujer A. L., Camargo F. and de Lima H. F.,Complete spacelike hypersurfaces with constant mean curvature in−R×Hn. J. Math. Anal. Appl. 368(2010), 650-657. · Zbl 1193.53124
[7] [7]Al´ıas L. J., Dajczer M., and Ripoll J. B.,A Bernstein-type theorem for Riemannian manifolds with a Killing field. Ann. Glob. Anal. Geom.31 (2007), 363-373.
[8] [8]Aquino C. P., de Lima H. F. and Lima Jr. E. A.,Complete CMC spacelike hypersurfaces immersed in a Lorentzian product space. Arch. Math.104 (2015), 577-587. · Zbl 1330.53076
[9] [9]Bartnik R.,Existence of maximal surfaces in asymptotically flat spacetimes. Comm. Math. Phys.94(1984), 155-175. · Zbl 0548.53054
[10] [10]Brendle S.,Constant mean curvature surfaces in warped product manifolds. Publ. Math. de l’IHES117(2013), 247-269. · Zbl 1273.53052
[11] [11]Calabi E.,Examples of Bernstein problems for some nonlinear equations. · Zbl 0211.12801
[12] [29]Treibergs A. E.,Entire Spacelike Hypersurfaces of Constant Mean Curvature in Minkowski Space. Invent. Math.66(1982), 39-56. · Zbl 0483.53055
[13] [30]Yau S. T.,Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math.28(1975), 201-228. · Zbl 0291.31002
[14] [31]Yau S.
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