Contracting an axially symmetric torus by its harmonic mean curvature

Christopher Kim

Vol. 268 (2014), No. 1, 117–133

DOI: 10.2140/pjm.2014.268.117

Abstract

We consider the harmonic mean curvature flow of an axially symmetric torus whose axis is a closed geodesic, where the ambient space is a hyperbolic three-manifold. Assuming the initial surface is strictly convex and its harmonic mean curvature is less than $\frac{1}{2}$, we show that the evolving surface satisfies a curvature condition comparable to that of a perfectly symmetric torus evolving under harmonic mean curvature flow. In other words, we prove that ${\lambda }_1\approx e^{-t}$, ${\lambda }_2\approx e^t$ and ${\lambda }_1{\lambda }_2\approx 1$, where ${\lambda }_1$ and ${\lambda }_2$ are the principal curvatures of the evolving torus.

Keywords

harmonic mean curvature flow, hyperbolic manifold, closed geodesic

Mathematical Subject Classification 2010

Primary: 53C44

Secondary: 35K55

Milestones

Received: 23 May 2013

Accepted: 30 December 2013

Published: 21 May 2014