Geometric expansion of starshaped plane curves
Geometric expansion of starshaped plane curves
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DOI：
10.4310/cag.1996.v4.n3.a5
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发表时间：
1996
影响因子：
0.7
通讯作者：
Dong-Ho Tsai
中科院分区：
文献类型：
--
作者：
Dong-Ho Tsai
在本文中，我们将处理 R 中任意星形闭合平面曲线 70 的一般非齐次扩展曲率流。这是 Chow-Tsai [5] 对闭合凸平面曲线非齐次扩展曲率流的部分推广。这两篇论文之间的主要区别在于，在 Chow-Tsai [5] 中，我们使用凸闭合曲线的支持函数来表述我们的问题，而这里我们使用星形闭合曲线的径向函数。在 Chow-LiouTsai [8] 的后续论文中，我们将把我们的结果推广到转角大于 —TT 的嵌入平面曲线的展开。虽然我们在这里关注的是非齐次扩展曲率流，但我们应该提到过去几年关于闭凸曲线和超曲面的齐次曲率流的许多重要工作。我们建议读者参阅 Andrews [1]、Chow-Tsai [5] 或 Chow-Liou-Tsai [7] 论文中的介绍性评论，以获取更多信息。对于扩展星形超曲面，我们提到了 Urbas [19] 和 Gerhardt [10] 的论文，他们考虑了维度至少为 2 的星形超曲面的均匀扩展曲率流。最近，Chow [2] 能够使用 Aleksandrov 反射方法获得星形超曲面非均匀扩展曲率流的径向函数的梯度估计。令 70 为任意星形（相对于 R 的原点）嵌入闭合曲线，由平滑嵌入 Xo(u) : S —> R 参数化。我们考虑初始值问题
In this paper we shall deal with general nonhomogeneous expanding curvature flows of an arbitrary starshaped closed plane curve 70 in R. This is a partial generalization of Chow-Tsai [5] for nonhomogeneous expanding curvature flows of closed convex plane curves. A major difference between these two papers is that in Chow-Tsai [5] we use the support functions of convex closed curves to formulate our problem while we use the radial functions of starshaped closed curves here. In a later paper by Chow-LiouTsai [8], we shall generalize our results to the expansion of embedded plane curves with turning angle greater than —TT. Although we concentrate on nonhomogeneous expanding curvature flows here, we should mention the many important works on homogeneous curvature flows of closed convex curves and hypersurfaces in the last several years. We refer the readers to the introductory remarks in the papers by Andrews [1], Chow-Tsai [5], or Chow-Liou-Tsai [7] for further information. For expanding starshaped hypersurfaces, we mention the papers by Urbas [19] and Gerhardt [10], where they consider the homogeneous expanding curvature flows of starshaped hypersurfaces with dimension at least two. Recently, Chow [2] was able to use the Aleksandrov reflection method to get a gradient estimate for the radial functions of nonhomogeneous expanding curvature flows of starshaped hypersurfaces. Let 70 be an arbitrary starshaped (with respect to the origin of R) embedded closed curve parameterized by a smooth embedding Xo(u) : S —> R. We consider the initial value problem