Inverse anisotropic mean curvature flow and a Minkowski type inequality

TL;DR

This paper studies the inverse anisotropic mean curvature flow starting from star-shaped hypersurfaces, proving long-time existence, exponential convergence to a Wulff shape, and establishing a Minkowski type inequality for such hypersurfaces.

Contribution

It introduces the analysis of inverse anisotropic mean curvature flow in $ eal^{n+1}$ and proves convergence to Wulff shapes along with a new Minkowski inequality.

Findings

Flow exists for all time from star-shaped, $F$-mean convex hypersurfaces.

Flow converges exponentially fast to a Wulff shape.

Establishes a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.

Abstract

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we prove a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.