Locally constrained inverse curvature flows

TL;DR

This paper studies inverse curvature flows in warped product manifolds with local constraints, proving long-term existence and convergence, and deriving new geometric inequalities in specific manifolds.

Contribution

It introduces a novel analysis of inverse curvature flows with local constraints and applies it to establish new Minkowski and isoperimetric inequalities.

Findings

Proved long-time existence and smooth convergence of constrained inverse curvature flows.

Derived a new Minkowski type inequality in anti-de-Sitter Schwarzschild manifolds.

Established a weighted isoperimetric inequality in hyperbolic space.

Abstract

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski type inequality in the anti-de-Sitter Schwarzschild manifolds and a weighted isoperimetric type inequality in the hyperbolic space.