曲线收缩流compactness及其与标度流形的关系
The curve shrinking flow, compactness and its relation to scale manifolds
摘要：
这篇硕士论文着眼于嵌入循环上的长度泛函的梯度流。嵌入环空间被赋予尺度结构，使得长度泛函成为尺度光滑的。对于某些基本流形，利用长度的梯度流与曲率流相同的事实，证明了固定临界点（测地线）之间的梯度流线的弱紧性质.还实现了某些梯度流线的Floer-Gromov收敛。这些收敛性质与莫尔斯理论中的收敛性质非常相似。图可视化的梯度/曲率流在几个歧管包括在内。
英文摘要：
This master thesis looks at the gradient flow of the length functional on embedded loops. The space of embedded loops is endowed with a scale structure so that the length functional becomes scale smooth. For certain underlying manifolds, using the fact that the gradient flow of the length is the same as the curvature flow, a weak compactness property for the gradient flow lines between fixed critical points (geodesics) is proven. Also Floer-Gromov convergence of certain gradient flow lines is achieved. These convergence properties are very similar to those in Morse theory. Figures visualizing the gradient/curvature flow on several manifolds are included.
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