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Bernstein type properties of two-sided hypersurfaces immersed in a Killing warped product

Antonio W. Cunha;  Eudes L. de Lima;  Henrique F. de Lima;  Eraldo A. Lima Jr.;  Adriano A. Medeiros — 2016

Studia Mathematica

Our purpose is to apply suitable maximum principles in order to obtain Bernstein type properties for two-sided hypersurfaces immersed with constant mean curvature in a Killing warped product M ⁿ × ρ ℝ , whose curvature of the base Mⁿ satisfies certain constraints and whose warping function ρ is concave on Mⁿ. For this, we study situations in which these hypersurfaces are supposed to be either parabolic, stochastically complete or, in a more general setting, L¹-Liouville. Rigidity results related to entire...

Parabolicity and rigidity of spacelike hypersurfaces immersed in a Lorentzian Killing warped product

Eudes L. de Lima;  Henrique F. de Lima;  Eraldo A. Jr. Lima;  Adriano A. Medeiros — 2017

Commentationes Mathematicae Universitatis Carolinae

In this paper, we extend a technique due to Romero et al. establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed into a Lorentzian Killing warped product whose Riemannian base has parabolic universal Riemannian covering. As applications, we obtain rigidity results concerning these hypersurfaces. A particular study of entire Killing graphs is also made.

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