Rigidity for nearly umbilical hypersurfaces in space forms

Abstract

Perez proved some $L^2$ inequalities for closed convex hypersurfaces immersed in the Euclidean space ${\mathbb{R}}^{n+1}$, more generally, for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is ${\mathbb{R}}^{n+1}$, the hyperbolic space ${\mathbb{H}}^{n+1}$, or the closed hemisphere ${\mathbb{S}}_{+}^{n+1}$. We also obtain a generalization of the Perez's theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.

Cited by in corpus (1)

• On an inequality of Andrews, De Lellis and Topping