SINGULARITIES OF MEAN CURVATURE FLOW AND ISOPERIMETRIC INEQUALITIES IN H3

In this paper, we mainly consider the mean curvature flow of surfaces in hyperbolic 3-space. First, we establish the isoperimetric inequality using the flow, provided the enclosed volume approaches zero at the final time. Second, we construct two singular examples of the flow. More precisely, there exists a torus which must develop a singularity in the flow before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of dumbbell, which must develop a singularity in the flow before its area shrinks to zero.