Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert space

TL;DR

This paper studies how certain invariant hypersurfaces in a Hilbert space evolve under mean curvature flow, showing they collapse to a group orbit if they meet specific convexity conditions.

Contribution

It demonstrates the collapse behavior of invariant hypersurfaces under regularized mean curvature flow in infinite-dimensional Hilbert spaces with group symmetries, under convexity assumptions.

Findings

Hypersurfaces satisfying convexity conditions collapse to group orbits.

Flow preserves invariance under the group action.

Collapse occurs along the regularized mean curvature flow.

Abstract

In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow.