Motion Of Level Set By General Curvature

TL;DR

This paper investigates the motion of level sets driven by general curvature functions, introducing new approximation techniques and an elliptic approach to extend weak solutions and non-collapsing results in nonlinear curvature flows.

Contribution

It introduces a novel approximation function to extend weak solutions beyond the admissible cone and applies an elliptic approach to non-collapsing results in curvature flows.

Findings

Extended existence of weak solutions outside the admissible cone

Developed a new approximation function for general curvature flows

Provided an elliptic approach to non-collapsing results

Abstract

In this paper, we study the motion of level sets by general curvature. The difficulty of this setting is that a general curvature function is only well defined in an admissible cone. In order to extend the existence of a weak solution of a general curvature flow to outside the cone we introduce a new approximation function $\hat{f}^n$ (see (3.1)). Moreover, using the idea in [4], we give an elliptic approach for the Ben-Andrews' non-collapsing result in fully nonlinear curvature flows; we hope this approach can be generalized to a wider class of elliptic equations.