Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

Article type: Research Article

Authors: Antonia, Railane^a | Molica Bisci, Giovanni^{b; *} | de Lima, Henrique F.^c | Santos, Márcio S.^a

Affiliations: [a] Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900 João Pessoa, Paraíba, Brazil | [b] Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy | [c] Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil

Correspondence: [*] Corresponding author. E-mail: giovanni.molicabisci@uniurb.it.

Abstract: We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Keywords: Semi-Riemannian warped products, complete hypersurfaces, higher order mean curvatures, entire graphs, maximum principles

DOI: 10.3233/ASY-231858

Journal: Asymptotic Analysis, vol. 135, no. 3-4, pp. 363-398, 2023