Abstract
In this paper, we extend a technique due to Romero et al. (Class Quantum Gravity 30:1–13, 2013; Int J Geom Methods Mod Phys 10:1360014, 2013; J Math Anal Appl 419:355–372, 2014) establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime whose fiber has \(\phi \)-parabolic universal Riemannian covering. As some applications of this criteria, we obtain uniqueness results concerning spacelike hypersurfaces immersed in spatially weighted generalized Robertson–Walker spacetimes. Furthermore, Calabi–Bernstein type results are also given.
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Aiyama, R.: On the Gauss map of complete space-like hypersurfaces of constant mean curvature in Minkowski space. Tsukuba J. Math. 16, 353–361 (1992)
Albujer, A.L., Alías, L.J.: Spacelike hypersurfaces with constant mean curvature in the steady state space. Proc. Am. Math. Soc. 137, 711–721 (2009)
Albujer, A.L., Camargo, F.E.C., de Lima, H.F.: Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Math. Proc. Camb. Philos. Soc. 151, 271–282 (2011)
Albujer, A.L., de Lima, H.F., Oliveira, A.M., Velásquez, M.A.L.: Rigidity of spacelike hypersurfaces in spatially weighted generalized Robertson–Walker spacetimes. Differ. Geom. Appl. 50, 140–154 (2017)
Alías, L.J., Colares, A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. Math. Proc. Camb. Philos. Soc. 143, 703–729 (2007)
Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relativ. Gravit. 27, 71–84 (1995)
An, H.V.Q., Cuong, D.V., Duyenb, N.T.M., Hieub, D.T., Nam, T.L.: On entire \(f\)-maximal graphs in the Lorentzian product \(\mathbb{G}^n\times \mathbb{R}_1\). J. Geom. Phys. 114, 587–592 (2017)
Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/1984. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)
Case, J.S.: Singularity theorems and the Lorentzian splitting theorem for the Bakry–Émery–Ricci tensor. J. Geom. Phys. 60, 477–490 (2010)
Cavalcante, M.P., de Lima, H.F., Santos, M.S.: New Calabi–Bernstein type results in weighted generalized Robertson–Walker spacetimes. Acta Math. Hung. 145, 440–454 (2015)
Colares, A.G., de Lima, H.F.: On the rigidity of spacelike hypersurfaces immersed in the steady state space \({\cal{H}}^{n+1}\). Publ. Math. Debr. 81, 103–119 (2012)
de Lima, H.F., Parente, U.L.: On the geometry of maximal spacelike hypersurfaces in generalized Robertson–Walker spacetimes. Ann. Mat. Pura Appl. 192, 649–663 (2013)
Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)
Grigor’yan, A.: Escape rate of Brownian motion on Riemannian manifolds. Appl. Anal. 71(1–4), 63–89 (1999)
Grigor’yan, A., Saloff-Coste, L.: Dirichlet heat-kernel in the exterior of a compact set. Commun. Pure Appl. Math. 55, 93–133 (2002)
Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hieu, D.T., Nam, T.L.: Bernstein type theorem for entire weighted minimal graphs in \(\mathbb{G}^n\times \mathbb{R}\). J. Geom. Phys. 81, 87–91 (2014)
Impera, D., Rimoldi, M.: Stability properties and topology at infinity of \(f\)-minimal hypersurfaces. Geom. Dedicata 178, 21–47 (2015)
Kanai, M.: Rough isometries and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Jpn. 37, 391–413 (1985)
Li, P.: Curvature and Function Theory on Riemannian Manifolds. Surveys in Differential Geometry, vol. VII. International Press, Cambridge, pp. 375–432 (2000)
Marsden, J.E., Tipler, F.J.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep. 66, 109–139 (1980)
Montiel, S.: Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314, 529–553 (1999)
Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Romero, A., Rubio, R.M., Salamanca, J.J.: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes. Class. Quantum Gravity 30, 1–13 (2013)
Romero, A., Rubio, R.M., Salamanca, J.J.: Parabolicity of spacelike hypersurfaces in generalized Robertson–Walker spacetimes. Applications to uniqueness results. Int. J. Geom. Methods Mod. Phys. 10, 1360014 (2013)
Romero, A., Rubio, R.M., Salamanca, J.J.: A new approach for uniqueness of complete maximal hypersurfaces in spatially parabolic GRW spacetimes. J. Math. Anal. Appl. 419, 355–372 (2014)
Rimoldi, M.: Rigidity results for Lichnerowicz Bakry–Émery Ricci tensors. Ph.D. thesis, Università degli Studi di Milano, Milano (2011)
Stumbles, S.M.: Hypersurfaces of constant mean extrinsic curvature. Ann. Phys. 133, 28–56 (1981)
Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Springer, Dordrecht (1994)
Wei, G., Wylie, W.: Comparison geometry for the Bakry–Émery–Ricci tensor. J. Differ. Geom. 83, 377–405 (2009)
Xin, Y.L.: On the Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space. Comment. Math. Helv. 66, 590–598 (1991)
Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
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Albujer, A.L., de Lima, H.F., Oliveira, A.M. et al. \(\phi \)-Parabolicity and the Uniqueness of Spacelike Hypersurfaces Immersed in a Spatially Weighted GRW Spacetime. Mediterr. J. Math. 15, 84 (2018). https://doi.org/10.1007/s00009-018-1134-8
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DOI: https://doi.org/10.1007/s00009-018-1134-8
Keywords
- Spatially weighted generalized Robertson–Walker spacetimes
- Bakry–Émery–Ricci tensor
- Drifted Laplacian
- \(\phi \)-Parabolicity
- Weighted mean curvature
- Complete spacelike hypersurfaces
- Entire vertical graphs