The normalized curve shortening flow and homothetic solutions
The normalized curve shortening flow and homothetic solutions
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DOI：
10.4310/jdg/1214440025
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发表时间：
1986
影响因子：
2.5
通讯作者：
U. Abresch;J. Langer
中科院分区：
文献类型：
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作者：
U. Abresch;J. Langer
曲线缩短问题，现在广为人知，是理解规则闭合曲线γ: R/Z -> M按照曲率法向量：dy/dt = kN = -“弧长ZΛgradient”运动的演化。这个问题的一个动机是C. Croke， H. Gluck， W. Ziller和其他人在这方面表达的观点：改进封闭大地测量理论中用于迭代缩短曲线的一些复杂和特别的结构是可取的。作为一个测试案例，它的目标是证明kN在平面上的简单封闭曲线空间上产生流的猜想，保持嵌入性并使这些曲线在长度接近零时渐近圆形。然而，γ曲率的演化方程是相当微妙的，这个猜想还没有得到解决。事实上，在非简单的情况下，人们通常期望单一的行为，而这个问题的部分内在兴趣在于，嵌入性的全局条件显然被“近视眼”方程所承认。到目前为止，我们所知道的是，这个猜想对凸曲线是正确的，简单曲线实际上仍然是简单的（如果曲率是有界的），并且方程的短时间解存在于完全的一般性中；这些结果归功于M. Gage和R. Hamilton（见[1]，[2],[3]）。
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient of arc length". One motivation for this problem has been the view expressed in this connection by C. Croke, H. Gluck, W. Ziller, and others: it would be desirable to improve on some complicated and ad hoc constructions that have been used in the theory of closed geodesies to iteratively shorten curves. As a test case it has been a goal to prove the conjecture that kN generates a flow on the space of simple closed curves in the plane, preserving embeddedness and making such curves circular asymptotically as length approaches zero. However, the evolution equation for the curvature of γ, turns out to be quite subtle, and the conjecture is not yet settled. Indeed, in the nonsimple case one generally expects singular behavior, and part of the intrinsic interest of the problem lies in the fact that the global condition of embeddedness is apparently recognized by the "near-sighted" equation. What is known thus far is that the conjecture is true for convex curves, that simple curves do in fact remain simple (provided curvature stays bounded), and that short time solutions to the equations exist in full generality; these results are due to M. Gage and R. Hamilton (see [1], [2], [3]).