论文标题
曲线缩短流的长期行为$ \ mathbb {r}^3 $
Long-term behavior of curve shortening flow in $\mathbb{R}^3$
论文作者
论文摘要
空间曲线运动描述了材料缺陷或接口的动力学，可以在图像处理或涡流动力学中找到。本文分析了曲线缩短流动的空间曲线的一些特性。与经典的平面曲线的经典案例相反，太空曲线一般不遵守回避原则。即使他们很简单，他们也可能会失去凸度或发展非圆形奇异性。在本文的第一部分中，我们表明，即使在运动过程中未保留太空曲线的凸度，它们的正交投影仍然是凸。在第二部分中，通过概括汉密尔顿和Gage开发的参数，显示了曲线缩短流下的球形曲线的回避原理。
Space curve motion describes dynamics of material defects or interfaces, can be found in image processing or vortex dynamics. This article analyses some properties of space curves evolved by the curve shortening flow. In contrast to the classical case of shrinking planar curves, space curves do not obey the Avoidance principle in general. They can lose their convexity or develop non-circular singularities even if they are simple. In the first part of the text, we show that even though the convexity of space curves is not preserved during the motion, their orthogonal projections remain convex. In the second part, the Avoidance principle for spherical curves under the curve shortening flow in $\mathbb{R}^3$ is shown by generalizing the arguments developed by Hamilton and Gage.
