On the variation of r-mean curvature functionals and application to the \(L^2\)-norm of the traceless second fundamental form

Abstract

Functionals involving surface curvatures are objects with applications in physics, mathematics, and related areas. It is then natural to study the minimizers of these functionals, as well as the stability of its critical points. In this paper, we begin by examining a general functional on n-dimensional hypersurfaces, which depends on the 1-mean curvature and the 2-mean curvature. We compute its first variation formula, obtaining the Euler-Lagrange equation that characterizes critical points. As a consequence, we also obtain the Euler-Lagrange equation for hypersurfaces immersed in Einstein manifolds as well as in manifolds with constant sectional curvature. In the case where the ambient space is a manifold with constant sectional curvature, we also compute the second variation, obtaining a stability criterion for these points in terms of geometric invariants that depend solely on the first and second fundamental forms. These results generalize those obtained in [7]. To demonstrate the applicability of the results, we studied the functional given by the \(L^2\)-norm of the traceless second fundamental form. From a geometric perspective, \(\Phi \) is a functional that measures how much M deviates from being totally umbilical, that is, from having equal principal curvatures at every point. We investigated the Euler-Lagrange equation and checked the stability of some known critical points.