Singularities of mean curvature flow and isoperimetric inequalities in H^3

Abstract

In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in ${\mathbb{H}}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of a dumbbell, which must develop a singularity in the flow before its area shrinks to zero. Simultaneously, by using the flow, we proved an isoperimetric inequality for some domains in ${\mathbb{H}}^3$.

arXiv comment12 pages

References in corpus (1)

• Non-collapsing for hypersurface flows in the sphere and hyperbolic space