Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

TL;DR

This paper investigates the volume-preserving mean curvature flow of revolution hypersurfaces in rotationally symmetric spaces, establishing conditions for long-term existence, convergence, and singularity structure.

Contribution

It provides new results on the existence, behavior, and convergence of hypersurfaces evolving under volume-preserving mean curvature flow in symmetric spaces.

Findings

Flow exists until the hypersurface touches the axis of rotation.

The generating curve remains a graph during evolution.

Under volume conditions, the flow converges to a constant mean curvature hypersurface.

Abstract

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the flow exists while M_t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M_t remains a graph, and b2) the averaged mean curvature is double side bounded by positive constants; c) the singularity set (if non-empty) is finite and discrete along the axis; d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.