On the extension of axially symmetric volume flow and 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.

arXiv commentarXiv admin note: text overlap with arXiv:1203.5671

References in corpus (2)

• The mean curvature at the first singular time of the mean curvature flow
• On the Convergence of Axially Symmetric Volume Preserving Mean Curvature Flow

Cited by in corpus (1)

• Analysis of Vel$\acute{a}$zquez's solution to the mean curvature flow with a type $\mathrm{II}$ singularity