Communications in Analysis and Geometry

Volume 30 (2022)

Number 8

Singularities of axially symmetric volume preserving mean curvature flow

Pages: 1683 – 1711



Maria Athanassenas (Defence Science and Technology Group. Eveleigh, New South Wales, Australia; and School of Mathematical Sciences, Monash University, Victoria, Australia)

Sevvandi Kandanaarachchi (CSIRO’s Data61, Clayton, Victoria, Australia)


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.

Received 18 January 2018

Accepted 5 March 2020

Published 13 July 2023