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


Author: Kui Wang
Journal: Proc. Amer. Math. Soc. 143 (2015), 2651-2660
MSC (2010): Primary 53C44; Secondary 52A40
Published electronically: February 5, 2015
MathSciNet review: 3326044
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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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Additional Information

Kui Wang
Affiliation: School of Mathematic Sciences, Fudan University, Shanghai, 200433, People’s Republic of China
Email: 09110180001@fudan.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2015-12490-2
Keywords: Mean curvature flow, isoperimetric inequality, Willmore energy, hyperbolic space
Received by editor(s): November 24, 2013
Published electronically: February 5, 2015
Additional Notes: The author was sponsored by the China Scholarship Council for two year study at University of California, San Diego.
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society