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

Longzhi Lin,Ling Xiao

Published 2010 in Communications in Analysis and Geometry

ABSTRACT

Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space Longzhi Lin Ling Xiao 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 intro- duced by De Silva and Spruck in [DS09]. 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 [GS00], and may be thought of as a parabolic analogue. Keywords. Modified mean curvature flow, Hyperbolic space, Star-shaped hypersurfaces Introduction Let F(z, t) : S n + × [0, ∞) → H n+1 be a one parameter family of complete embedded star-shaped hypersurfaces which are radial graphs in H n+1 over S n + , the upper hemisphere of the unit sphere S n in R n+1 , where the half-space model of H n+1 is used. We say the images Σ t = F(z, t) move by modified mean curvature flow (MMCF) if  ∂ F(z, t) ⊥ = (H − σ) ν , (z, t) ∈ S n × [0, ∞) , H ∂t F(z, 0) = Σ 0 , z ∈ S n + , where H denotes the hyperbolic mean curvature of Σ t , σ ∈ (−1, 1) is a constant, and ν H denotes the outward unit normal of Σ t with respect to the hyperbolic metric. By the half-space model of H n+1 , we mean H n+1 = {(x 0 , x n+1 ) ∈ R n+1 : x n+1 > 0} L. Lin: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA; e-mail: lzlin@math.rutgers.edu L. Xiao: Department of Mathematics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA; e-mail: lxiao@math.jhu.edu Mathematics Subject Classification (2010): Primary 53C44; Secondary 35K20, 58J35

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