Minkowski Inequality on complete Riemannian manifolds with nonnegative Ricci curvature

TL;DR

This paper proves an optimal Minkowski inequality for bounded domains in complete Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth, including characterizations of equality cases and related monotonicity formulas.

Contribution

It establishes the Minkowski inequality in a broad geometric setting and characterizes equality cases, extending classical results to noncompact manifolds with nonnegative Ricci curvature.

Findings

Proved Minkowski inequality for domains in manifolds with nonnegative Ricci curvature.

Characterized the equality case for strictly outward minimising and mean convex domains.

Established sharp monotonicity formulas along level sets of p-capacitary potentials.

Abstract

In this paper we consider Riemannian manifolds of dimension at least $3$, with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski Inequality. We also characterise the equality case, provided the domain is strictly outward minimising and strictly mean convex. Along with the proof, we establish in full generality sharp monotonicity formulas, holding along the level sets of $p$-capacitary potentials in $p$-nonparabolic manifolds with nonnegative Ricci curvature.