Singularities of Axially Symmetric Volume Preserving Mean Curvature Flow

TL;DR

This paper studies how axially symmetric surfaces evolving under volume-preserving mean curvature flow develop singularities, proving that the first singularity is of Type I without extra curvature assumptions, using advanced geometric estimates.

Contribution

It establishes that the first singularity in axially symmetric volume-preserving mean curvature flow is of Type I, under minimal assumptions, extending understanding of singularity formation.

Findings

First singularity is of Type I in axially symmetric flows.

Established maximum principle and geometric estimates in arbitrary dimensions.

Results hold without additional curvature assumptions.

Abstract

We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions are justifiable given the volume constraint. Additional results and ingredients towards the main proof include a non-cylindrical parabolic maximum principle, and a series of estimates on geometric quantities involving gradient, curvature terms and derivatives thereof. These hold in arbitrary dimensions.