Calculus of Variations and Geometric Measure Theory

V. Agostiniani - M. Fogagnolo - L. Mazzieri

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

created by fogagnolo on 21 Jun 2019
modified on 26 Oct 2020


Published Paper

Inserted: 21 jun 2019
Last Updated: 26 oct 2020

Journal: Inventiones Mathematicae
Year: 2018
Doi: 10.1007/s00222-020-00985-4

ArXiv: 1812.05022 PDF


In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial \Omega$ in $M$, with equality holding true if and only if $(M{\setminus}\Omega, g)$ is isometric to a truncated cone over $\partial\Omega$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.