Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms

Abstract

In Perez (Thesis, 2011), Perez proved some L2 inequalities for closed convex hypersurfaces immersed in the Euclidean space ℝn+1, and 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 ℝn+1, the hyperbolic space ℍn+1, or the closed hemisphere \(\mathbb{S}_{+}^{n+1}\). We also obtain a generalization of Perez’s theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.