Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants

TL;DR

This paper investigates the volume-preserving mean curvature flow of revolution hypersurfaces in symmetric spaces, proving long-term existence and convergence to constant mean curvature shapes under certain conditions.

Contribution

It extends the analysis of mean curvature flow to rotationally symmetric spaces with boundary conditions, including spaces with positive curvature, and establishes convergence results.

Findings

Flow exists as long as hypersurfaces do not touch the rotation axis.

Hypersurfaces converge to constant mean curvature shapes under volume-area conditions.

Results hold even in spaces with positive curvature.

Abstract

In a rotationally symmetric space $\oM$ around an axis A (whose precise definition includes all real space forms), we consider a domain $G$ limited by two equidistant hypersurfaces orthogonal to A. Let $M \subset \oM$ be a revolution hypersurface generated by a graph over A, with boundary in $\partial G$ and orthogonal to it. We study the evolution $M_t$ of $M$ under the volume-preserving mean curvature flow requiring that the boundary of $M_t$ rests on $\partial G$ and keeps orthogonal to it. We prove that: a) the generating curve of $M_t$ remains a graph; b) the flow exists while $M_t$ does not touch the axis of rotation; c) under a suitable hypothesis relating the enclosed volume and the area of $M$, the flow is defined for every $t\in [0,\infty[$ and a sequence of hypersurfaces $M_{t_n}$ converges to a revolution hypersurface of constant mean curvature. Some key points are: i) the results are true even for ambient spaces with positive curvature, ii) the averaged mean curvature does not need to be positive and iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.