Deforming a Convex Hypersurface by Anisotropic Curvature Flows

Hongjie Ju,Boya Li,Yannan Liu

doi:10.1515/ans-2020-2108

Abstract

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.

Highlights

Let M0 be a smooth, closed, strictly convex hypersurface in the Euclidean space Rn, which encloses the origin and is given by a smooth embedding X0 : Sn−1 → Rn
Principal radii of curvature are the eigenvalues of the matrix bij := ∇i∇jh + eijh with respect to {eij}. σk(x, t) is the k-th elementary symmetric function for principal curvature radii of Mt at X(x, t) and k is an integer with 1 ≤ k ≤ n − 1
We study the long-time existence and convergence of flow (1) for strictly convex hypersurfaces and the existence of smooth solutions to the Orlicz

Summary

Introduction

Let M0 be a smooth, closed, strictly convex hypersurface in the Euclidean space Rn, which encloses the origin and is given by a smooth embedding X0 : Sn−1 → Rn. We study the long-time existence and convergence of flow (1) for strictly convex hypersurfaces and the existence of smooth solutions to the Orlicz-. When t → ∞, a subsequence of Mt converges in C∞ to a smooth, closed, strictly convex hypersurface, whose support function is a smooth solution to equation (2) for some positive constant c.

Preliminaries

Findings

The long-time existence of the flow