Deforming a convex hypersurface by anisotropic curvature flows

TL;DR

This paper studies a nonlinear curvature flow of convex hypersurfaces involving symmetric functions of principal curvatures and support functions, proving long-term existence, convergence, and solving a related geometric problem.

Contribution

It introduces a new curvature flow involving elementary symmetric functions and support functions, establishing long-term behavior and solving the Orlicz-Christoffel-Minkowski problem.

Findings

Proves long-time existence of the flow

Shows convergence of the flow to a smooth limit

Establishes existence of solutions to the Orlicz-Christoffel-Minkowski problem

Abstract

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem.