Stability and bifurcation of equilibria for the axisymmetric averaged mean curvature flow

TL;DR

This paper analyzes the stability and bifurcation behavior of axisymmetric surfaces under volume-preserving mean curvature flow, establishing well-posedness and dynamic properties of equilibria including cylinders.

Contribution

It provides the first analytic well-posedness results for the flow with rough initial data and characterizes bifurcation phenomena of cylindrical equilibria.

Findings

Well-posedness of the flow in little-Hölder spaces

Stability and instability regions for cylindrical equilibria

Bifurcation behavior depending on cylinder radius

Abstract

We study the averaged mean curvature flow, also called the volume preserving mean curvature flow, in the particular setting of axisymmetric surfaces embedded in R^3 satisfying periodic boundary conditions. We establish analytic well--posedness of the flow within the space of little-H\"older continuous surfaces, given rough initial data. We also establish dynamic properties of equilibria, including stability, instability, and bifurcation behavior of cylinders, where the radius acts as a bifurcation parameter.