Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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DEFORMING PINCHED HYPERSURFACES OF THE HYPERBOLIC SPACE BY POWERS OF THE MEAN CURVATURE INTO SPHERES

  • Guo, Shunzi (School of Mathematics Sichuan University, School of Mathematics and Statistics Minnan Normal University) ;
  • Li, Guanghan (School of Mathematics and Statistics Wuhan University) ;
  • Wu, Chuanxi (School of Mathematics and Computer Science Hubei University)
  • Received : 2014.07.22
  • Published : 2016.07.01
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Abstract

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ 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

Acknowledgement

Supported by : National Natural Science Foundation, China Postdoctoral Science Foundation, Natural Science Foundation

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