Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms

Abstract

In Perez (Thesis, 2011), Perez proved some L ² 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.