The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action

TL;DR

This paper investigates the behavior of the regularized mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action, establishing convexity preservation results.

Contribution

It introduces evolution equations for geometric quantities and proves convexity preservation theorems for the flow in the context of Hilbert spaces with group actions.

Findings

Convexity is preserved under the flow for invariant hypersurfaces.

Derived evolution equations for geometric quantities along the flow.

Established convexity preservation in the orbit space setting.

Abstract

In this paper, we study the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds, where "almost free" means that the stabilizers of the group action are finite. First we obtain the evolution equations for some geometric quantities along the regularized mean curvature flow. Next, by using the evolution equations, we prove a horizontally strongly convexity preservability theorem for the regularized mean curvature flow. From this theorem, we derive the strongly convexity preservability theorem for the mean curvature flow starting from compact Riemannian suborbifolds in the orbit space (which is a Riemannian orbifold) of the Hilbert Lie group action.