A New Monotone Quantity along the Inverse Mean Curvature Flow in $\mathbb R^n$

TL;DR

This paper introduces a new monotone quantity along the inverse mean curvature flow in Euclidean space and uses it to establish a sharp geometric inequality relating volume and mean curvature for certain hypersurfaces.

Contribution

The paper presents a novel monotone quantity for the inverse mean curvature flow and applies it to derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces.

Findings

Established a new monotone increasing quantity along the flow

Derived a sharp geometric inequality relating volume and mean curvature

Applied the monotone quantity to characterize geometric properties of hypersurfaces

Abstract

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypersurface.