Singularities of mean curvature flow and isoperimetric inequalities in ℍ³

Wang, Kui

doi:10.1090/S0002-9939-2015-12490-2

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

Proc. Amer. Math. Soc. 143 (2015), 2651-2660

DOI: https://doi.org/10.1090/S0002-9939-2015-12490-2

Published electronically: February 5, 2015

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Abstract:

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.

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