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

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 ${\mathbb{R}}^3$ satisfying periodic boundary conditions. We establish analytic well-posedness of the flow within the space of little-Hölder 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.