Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

Details Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

2010-10-11

arxiv.org/abs/1010.2091v1

Mathematics

Differential Geometry

26 pages, 3 figures

Scientific paper

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, which was first shown by Guan and Spruck.

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