A new geometric flow with rotational invariance

TL;DR

This paper introduces a novel geometric flow with rotational invariance that guarantees convergence of smooth closed hypersurfaces in Euclidean space to spheres, extending classical results in curvature flows.

Contribution

The paper presents a new rotationally invariant geometric flow and proves its convergence properties, generalizing classical curvature flow theorems.

Findings

Hypersurfaces converge to spheres under the new flow

The flow generalizes Gage-Hamilton and Huisken's results

Convergence occurs in the C1-topology as time approaches infinity

Abstract

In this paper we introduce a new geometric flow with rotational invariance and prove that, under this kind of flow, an arbitrary smooth closed contractible hypersurface in the Euclidean space Rn+1 (n, 1) converges to Sn in the C1-topology as t goes to the infinity. This result covers the well-known theorem of Gage and Hamilton in [4] for the curvature flow of plane curves and the famous result of Huisken in [5] on the flow by mean curvature of convex surfaces, respectively.