Is Mean Curvature Flow a Gradient Flow?

TL;DR

This paper investigates whether the mean curvature flow can be represented as a gradient flow in nondegenerate metric spaces of plane curves, concluding that it cannot in the studied cases.

Contribution

It demonstrates that the mean curvature flow does not admit a gradient flow structure in two specific nondegenerate metric spaces of plane curves.

Findings

Mean curvature flow is not a gradient flow in the uniformness-preserving metric space.

Mean curvature flow is not a gradient flow in the curvature-weighted metric space.

The study clarifies limitations of gradient flow representations for mean curvature flow.

Abstract

It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces.