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

Abstract

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension $$n \ge 3$$ . For every bounded open subset $$\Omega \subset M$$ with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\,\big |\mathbb {S}^{n-1}\big |, \end{aligned}$$ where $${\mathrm{H}}$$ is the mean curvature of $$\partial \Omega $$ and $${\mathrm{AVR}}(g)$$ is the asymptotic volume ratio of (M, g). Moreover, the equality holds 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.