Uniformly compressing mean curvature flow

TL;DR

This paper introduces a new geometric flow related to mean curvature flow that avoids certain drawbacks, with analytical results on well-posedness and stability in the 1D case.

Contribution

It proposes a novel flow as a gradient flow on a submanifold of Wasserstein space, addressing issues of uniform density destruction and degeneracy.

Findings

Flow is well-posed in 1D

Long-time stability established in 1D

Avoids destruction of uniform density

Abstract

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of the Wasserstein space of probability measures endowed with Otto's Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are however restricted to the 1D case of evolution of loops.