On the extension of axially symmetric volume flow and mean curvature flow

TL;DR

This paper investigates conditions preventing singularity formation in axially symmetric volume and mean curvature flows, showing that bounded mean curvature ensures smooth evolution without singularities.

Contribution

It establishes that uniform boundedness of mean curvature guarantees no singularities develop in axially symmetric volume and mean curvature flows.

Findings

Bounded mean curvature prevents singularities in the flows.

No singularities develop if mean curvature remains uniformly bounded.

Results apply to both mean curvature flow and volume-preserving mean curvature flow.

Abstract

We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow and volume preserving mean curvature flow.