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

## TL;DR

This paper establishes sharp geometric inequalities for closed hypersurfaces in noncompact manifolds with nonnegative Ricci curvature, leading to new isoperimetric bounds and non-existence results for minimal hypersurfaces.

## Contribution

It introduces a sharp Willmore-type inequality and an optimal Huisken's isoperimetric inequality for such manifolds, extending to parabolic cases and improving existing non-existence results.

## Key findings

- Proves a sharp Willmore-type inequality with equality characterizations.
- Derives an optimal Huisken's isoperimetric inequality for 3-manifolds.
- Extends techniques to parabolic manifolds to improve Kasue's non-existence results.

## Abstract

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.