Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
复制标题
DOI：
10.1002/mana.201700370
复制
发表时间：
2017-03
影响因子：
1
通讯作者：
H. Kröner;Julian Scheuer
中科院分区：
文献类型：
--
作者：
H. Kröner;Julian Scheuer
证明了扩展曲率流在欧氏空间和双曲空间中的收敛性结果。流速有F−p的形式，其中p>1和F是一个正的，严格单调的和1‐齐次的曲率函数。特别地，这一类包括平均曲率F=H。我们证明了保留了一定的初始捏紧条件，并且适当缩放的超曲面平滑地收敛到单位球上。我们证明了Andrews-McCoy-Zheng的一个例子可以用来构造严格凸的初始超曲面，其中逆平均曲率流的p>次方失去了凸性，证明了在初始超曲面上施加一定的捏紧条件的必要性。
We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form F−p , where p>1 and F is a positive, strictly monotone and 1‐homogeneous curvature function. In particular this class includes the mean curvature F=H . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews–McCoy–Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power p>1 loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.