A Minkowski type inequality for manifolds with positive spectrum

TL;DR

This paper extends Minkowski inequalities to non-convex domains in manifolds with large bottom spectrum, leading to nonexistence results for minimal hypersurfaces and applications to Ricci solitons.

Contribution

It establishes a Minkowski type inequality for non-convex domains in manifolds with large bottom spectrum, generalizing classical results.

Findings

Minkowski inequality holds without convexity in certain manifolds

Nonexistence of embedded compact minimal hypersurfaces in these manifolds

Applications to steady and expanding Ricci solitons

Abstract

The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity assumption for all bounded smooth domains in a complete manifold with its bottom spectrum being suitably large relative to its Ricci curvature lower bound. An immediate implication is the nonexistence of embedded compact minimal hypersurfaces in such manifolds. This nonexistence issue is also considered for steady and expanding Ricci solitons.