J. Korean Math. Soc. 2016; 53(4): 737-767
Printed July 1, 2016
https://doi.org/10.4134/JKMS.j140445
Copyright © The Korean Mathematical Society.
Shunzi Guo, Guanghan Li, and Chuanxi Wu
Minnan Normal University, Wuhan University, Hubei University
This paper concerns closed hypersurfaces of dimension $n\geq 2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed equals a power $\beta \geq1$ of the mean curvature. The main result is that if the initial closed, weakly $h$-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to $1$, depending only on $n$ and $\beta$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.
Keywords: powers of the mean curvature, horosphere, convex hypersurface, hyperbolic space, normalization
MSC numbers: Primary 53C44, 35K55; Secondary 58J35, 35B40
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