Curvature flows in the sphere
Curvature flows in the sphere
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DOI：
10.4310/jdg/1430744123
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发表时间：
2013-08
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影响因子：
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通讯作者：
Claus Gerhardt
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文献类型：
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作者：
Claus Gerhardt
我们考虑收缩和扩张曲率流在$\Ss$。当流超曲面是严格凸的时，我们建立了收缩超曲面和扩张超曲面之间的关系，该关系由Gau映射给出.收缩的超曲面收缩到一个点x_0，而扩张的超曲面收敛到半球的赤道。在重新标度后，通过相同的标度因子，重新标度的超曲面在$C^\un（\Ss[n]）$中指数快速地收敛到中心为x_0 $ \resp $-x_0$的单位球面。
We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gau{\ss} map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersurfaces converge to the equator of the hemisphere $\mc H(-x_0)$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers $x_0$ \resp $-x_0$ exponentially fast in $C^\un(\Ss[n])$.