Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

TL;DR

This paper studies the volume-preserving mean curvature flow of tubes in rank one symmetric spaces of non-compact type, showing preservation of tubeness and convergence properties under certain conditions.

Contribution

It proves the preservation of tubeness and convergence of the flow to a constant mean curvature tube in specific symmetric spaces, extending understanding of geometric flows in these settings.

Findings

Tubeness is preserved along the flow in specified symmetric spaces.

The flow either reaches the invariant submanifold or converges to a constant mean curvature tube.

Convergence occurs in the $C^{ abla} topology in infinite time.

Abstract

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type (other than a (real) hyperbolic space) and the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature over the invariant submanifold in the $C^{\infty}$-topology in infinite time.