半球中的非齐次曲率流
Non-homogeneous curvature flows in a hemisphere
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中文总结 AI 辅助
研究半球中非齐次曲率流的长时间行为，在结构性凸条件下证明严格凸性保持，并给出归一化径向函数指数收敛及超曲面收缩至原点的结论。
中文摘要 AI 辅助
设S^{n+1}_{+}为以o为中心的单位球面S^{n+1}的开半球。我们研究光滑、封闭、严格凸且包围o的超曲面的非齐次曲率流X_t=-f(r) sigma_k^{alpha} {nu}，其中r是到o的测地距离。我们同时考虑超临界情形beta>1+k{alpha}和临界情形beta=1+k{alpha}，其中beta是剖面在原点处的增长阶，f(r)在r趋于0时几乎等于r^{beta}。在f^{1/(1+k{alpha})}为凸的结构性条件下，我们证明长时间存在性和严格凸性的保持。归一化径向函数光滑且指数地收敛到一个常数：在两种不同情形下分别收敛到1和R_{infty}>0。因此归一化径向图变为圆形，而原始超曲面收缩到o。
英文摘要
Let S^{n+1}_{+} be the open hemisphere of the unit sphere S^{n+1} centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k {alpha} and the critical regime beta=1+k{alpha}, where beta is the growth order of the profile at the origin, f(r) almost equals r^{beta} as r descends to 0. Under the structural condition that f^{1/(1+k{alpha})} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_{infty}>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.
发表机构
- Westlake University（西湖大学）
- Zhejiang University（浙江大学）
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