Curvature flow in hyperbolic spaces

Abstract

We study the evolution of compact convex hypersurfaces in hyperbolic space H nC1 , with normal speed governed by the curvature. We concentrate mostly on the case of surfaces, and show that under a large class of natural flows, any compact initial surface with Gauss curvature greater than 1 produces a solution which converges to a point in finite time, and becomes spherical as the final time is approached. We also consider the higher-dimensional case, and show that under the mean curvature flow a similar result holds if the initial hypersurface is compact with positive Ricci curvature.

Self-similar solutions of curvature flows in warped products

“…In 3-dimensional hyperbolic space H 3 , deforming surfaces by a speed function σ 2 − 1 is studied in [3]. For self-similar solutions to a relevant curvature flow in H 3 , we obtain the following theorem.…”

Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds

“…Curvature flow in more complex ambient spaces has been treated analytically in e.g. [1,20,37], while numerical approximations have been considered in [6,12,21,55,56,60], for the case of closed curves, and in [16] for the case of open curves. Using the flow along the negative gradient of the total squared geodesic curvature functional to obtain geodesics as long-time limits has been first suggested in a seminal paper by Langer and Singer, [51].…”

Singularities of mean curvature flow and isoperimetric inequalities in $\mathbb {H}^3$

“…The flow (1.1) has been a useful topic in the study of geometry problems and there are many good results of the flow, beginning with the G. Huisken's paper [11] in the case M n = R n . They proved in [1] that the flow Σ t with initial Gauss curvature GK > 1 shrinks to a point and becomes more spherical. They proved in [1] that the flow Σ t with initial Gauss curvature GK > 1 shrinks to a point and becomes more spherical.…”

“…Thus the condition GK > 1 in [1] may not be replaced by the integration condition(1.5). Thus the condition GK > 1 in [1] may not be replaced by the integration condition(1.5).…”