Volume-Preserving Mean Curvature Flow for Tubes in Rank One Symmetric Spaces of Non-compact Type

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 closed geodesic ball in an invariant submanifold in a rank one symmetric space of non-compact type, where we impose some boundary condition to the flow and the invariancy of the submanifold means the total geodesicness in the case where the ambient symmetric space is a (real) hyperbolic space. Next, we prove that the tubeness is preserved along the flow in the case where the radius function of the initial tube is radial with respect to the center of the closed geodesic ball. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to a tube of constant mean curvature over the closed geodesic ball in the $C^{\infty }$-topology in infinite time.