Contraction of convex hypersurfaces in Riemannian spaces
Contraction of convex hypersurfaces in Riemannian spaces
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DOI：
10.4310/jdg/1214454878
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发表时间：
1994
影响因子：
2.5
通讯作者：
B. Andrews
中科院分区：
文献类型：
--
作者：
B. Andrews
本文利用Weingarten曲率定义的完全非线性抛物方程，讨论了黎曼空间中超曲面的变形问题。结果表明，任何满足自然凸性条件的初始超曲面都会产生一个在有限时间内收敛到单点的解，并且随着极限的逼近而变成球面。这一结果具有拓扑学意义，包括对Klingenberg，Berger和Rauch的1/4-Pinching球定理的新证明，以及允许某些负曲率的新的“凹陷球定理”。
This paper concerns the deformation of hypersurfaces in Riemannian spaces using fully nonlinear parabolic equations defined in terms of the Weingarten curvature. It is shown that any initial hypersurface satisfying a natural convexity condition produces a solution which converges to a single point in finite time, and becomes spherical as the limit is approached. The result has topological implications including a new proof of the 1/4-pinching sphere theorem of Klingenberg, Berger, and Rauch, and a new "dented sphere theorem" which allows some negative curvature.