Minkowski inequalities and constrained inverse curvature flows in warped spaces

TL;DR

This paper studies inverse curvature flows in warped spaces, proving long-term existence and convergence, and deriving new Minkowski inequalities in various dimensions based on curvature conditions.

Contribution

It introduces new Minkowski inequalities derived from inverse curvature flows in warped spaces, extending results to higher dimensions with specific curvature assumptions.

Findings

Flows converge smoothly to radial slices

New Minkowski inequalities are established

Inverse mean curvature flow yields results in negatively curved spaces

Abstract

This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows we prove long time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases new Minkowski type inequalities are the consequence. In higher dimensions we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.