Contracting convex torus by its harmonic mean curvature flow

Abstract

We consider the problem of contracting convex torus in Hyperbolic space by the harmonic mean of the principal curvatures. The shape of the torus is studied theoretically and numerically as each point on the torus moves towards the axis with a speed equal to the harmonic mean curvature. Due to the contrasting behavior \lambda_{1} \to 0, \lambda_{2} \to \infty of the principal curvatures of contracting torus, HMCF of torus is expected to be uniformly parabolic in \lambda_{1}-direction but degenerating in \lambda_{2}-direction. For the theoretical part, we assume the torus is axially symmetric and obtain estimates of the gradient function and the harmonic mean curvature using the parabolic maximum principle. The main result is that \lambda_{1} \approx e^{-t}, \lambda_{2} \approx e^{t}, \lambda_{1}\lambda_{2} \approx 1. We verify that HMCF is indeed uniformly parabolic and the shape in the limit is close to torus. We employ numerical methods to explore the case of general torus and provide numerical evidence that the torus does not evolve into a round shape if the initial surface has a low frequency component in \lambda_{2} -direction.