Global solutions to the volume-preserving mean-curvature flow

TL;DR

This paper develops a method to construct global solutions for the volume-preserving mean-curvature flow using a time-discrete gradient flow approach, advancing mathematical understanding of geometric evolution equations.

Contribution

It introduces a novel application of a time-discrete gradient flow method to establish global distributional solutions for the volume-preserving mean-curvature flow.

Findings

Successfully constructs global solutions in a distributional sense.

Extends the applicability of gradient flow methods to geometric flows.

Provides a framework for future analysis of volume-preserving curvature flows.

Abstract

In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren, Taylor and Wang (SIAM J. Control Optim. 31(2): 387- 438, 1993) and Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2): 253-271, 1995).