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2012•Communications in Analysis and GeometryOpen access

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

Longzhi Lin, Ling Xiao

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Abstract

We define a new modified mean curvature flow (MMCF) in hyperbolic space H n+1 , which interestingly turns out to be the natural negative L 2 -gradient flow of the energy functional introduced by De Silva and Spruck in [5].We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with prescribed asymptotic boundary at infinity.The proof of our main theorems follows closely Guan and Spruck's work [9], and may be thought of as a parabolic analog.

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We define a new modified mean curvature flow (MMCF) in hyperbolic space H n+1 , which interestingly turns out to be the natural negative L 2 -gradient flow of the energy functional introduced by De Silva and Spruck in [5].We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with prescribed asymptotic boundary at infinity.The proof of our main theorems follows closely Guan and Spruck's work [9], and may be thought of as a parabolic analog.

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Available abstract

We define a new modified mean curvature flow (MMCF) in hyperbolic space H n+1 , which interestingly turns out to be the natural negative L 2 -gradient flow of the energy functional introduced by De Silva and Spruck in [5].We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with prescribed asymptotic boundary at infinity.The proof of our main theorems follows closely Guan and Spruck's work [9], and may be thought of as a parabolic analog.

Key concepts: Mathematics, Mean curvature flow, Uniqueness, Hyperbolic space, Infinity, Mathematical analysis, Mean curvature, Boundary (topology)

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