The harmonic mean curvature flow of nonconvex surfaces in $\mathbb{R}^3$

TL;DR

This paper studies the harmonic mean curvature flow of nonconvex, star-shaped, mean convex surfaces in three-dimensional space, showing conditions under which the flow exists and shrinks to a point in a spherical manner.

Contribution

It extends understanding of curvature flows by analyzing nonconvex surfaces, proving existence and spherical shrinking behavior for specific classes of surfaces.

Findings

Flow exists until the surface shrinks to a point in some cases.

For surfaces of revolution, smooth solutions exist up to a finite time.

The flow causes surfaces to shrink asymptotically spherically.

Abstract

We consider a compact, star-shaped, mean convex hypersurface $\Sigma^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time $T < \infty$ at which the flow shrinks to a point asymptotically spherically.