A new characterization of $r$-stable hypersurfaces in space forms

H. F. de Lima and M. A. Velásquez

Address: Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, Paraíba, Brazil 58109-970


Abstract: In this paper we study the $r$-stability of closed hypersurfaces with constant $r$-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the $r$-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature.

AMSclassification: primary 53C42; secondary 53B30, 53C50, 53Z05, 83C99.

Keywords: space forms, $r$-th mean curvatures, $r$-stability.