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

TL;DR

This paper constructs examples of singularities in mean curvature flow within hyperbolic 3-space and establishes an isoperimetric inequality for certain domains, advancing understanding of geometric flows in hyperbolic geometry.

Contribution

It provides explicit singularity examples in MCF in H^3 and proves a new isoperimetric inequality using the flow method.

Findings

Existence of a torus developing singularity before volume vanishes

Existence of a dumbbell-shaped sphere developing singularity before area shrinks

Established an isoperimetric inequality for specific domains 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 $\mathbb{H}^3$.