Bifurcation and local rigidity of constant second mean curvature hypersurfaces in Riemannian warped products. (August 2020)

• Record Type: Journal Article
• Authors: Velásquez, Marco A.L.; Ramalho, André F.A.; da Silva, Jonatan F.; Oliveira, Jobson Q.
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: primary 53C42 -- secondary 58J55 35B32 35P15; Riemannian warped product -- H2-hypersurfaces -- Local rigidity -- Bifurcation instants -- Morse index; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111865
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In a Riemannian warped product I × f M n, where I ⊂ R is an open interval, f is a positive real function defined on I and M n is a compact Riemannian manifold without boundary, we use equivariant bifurcation theory in order to establish sufficient conditions, in terms of f and the spectrum of the Laplacian on M n, that allow us to guarantee the existence of bifurcation instants or the local rigidity of a certain family of open sets whose boundaries are H 2 -hypersurfaces, namely, whose boundaries are hypersurfaces with constant second mean curvature H 2 . For each of our results, we have provided a considerable number of examples that verify all the assumptions under consideration.