Communications in Analysis and Geometry

Volume 20 (2012)

Number 5

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

Pages: 1061 – 1096



Longzhi Lin (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Ling Xiao (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)


We define a new 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 introduced by De Silva and Spruck. 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, and may be thought of as a parabolic analog.

2010 Mathematics Subject Classification

Primary 53C44. Secondary 35K20, 58J35.

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