Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

TL;DR

This paper studies a curvature flow of hypersurfaces in Euclidean space that preserves mixed volume and shows that under certain conditions, the hypersurfaces evolve smoothly and exponentially approach a sphere.

Contribution

It generalizes previous results by establishing long-time existence and convergence for a broader class of curvature flows with mixed volume preservation.

Findings

Hypersurfaces converge exponentially to a sphere.

Flow exists smoothly for all time under pinching conditions.

Includes flows driven by powers of mean and Gauss curvature.

Abstract

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $\beta (\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy \cite{McC05} and Cabezas-Rivas and Sinestrari \cite{CS10} to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.