Curve-straightening and the Palais-Smale condition
Curve-straightening and the Palais-Smale condition
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曲线拉直和 Palais-Smale 条件
DOI：
10.1090/s0002-9947-98-01977-1
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发表时间：
1998
影响因子：
1.3
通讯作者：
Anders Linnér
中科院分区：
文献类型：
--
作者：
Anders Linnér
本文研究了修正的全平方曲率泛函fk ~ 2 + ivds的负梯度轨线。重点是当M从正侧趋于零时的限制行为。它表明，当M = 0的曲线空间存在，其中一些轨迹收敛和其他发散。在一个实例中，临界点的集合分成两个子集。当M趋于零时，第一子集中的临界曲线趋于当M 0时存在的临界点。同时，第二个子集中的所有临界点的长度趋于无穷大。它表明，这是唯一的方式的Palais-Smale条件失败，在目前的情况下。第二类临界点的行为支持这样的观点，即某些轨迹被“拖”到“无穷大”。当曲线被重新缩放为具有恒定长度时，欧拉8形作为“无穷大的临界点”出现。发现反射对称性不需要沿轨迹保持沿着。有例子表明，沿着沿着同一轨迹的曲线的长度，并不是流动时间的单调函数。它示出了如何确定在所有的标准情况下的临界曲线的椭圆模。模p必须满足2 E（p）/K（p）-1？gl/L，当空间限于固定长度L的曲线并且端点由向量g分隔时。
This paper considers the negative gradient trajectories associated with the modified total squared curvature functional f k2 + iv ds. The focus is on the limiting behavior as M tends to zero from the positive side. It is shown that when M = 0 spaces of curves exist in which some trajectories converge and others diverge. In one instance the collection of critical points splits into two subsets. As M tends to zero the critical curves in the first subset tend to the critical points present when M 0. Meanwhile, all the critical points in the second subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present context. The behavior of the second class of critical points supports the view that some of the trajectories are 'dragged' all the way to 'infinity'. When the curves are rescaled to have constant length the Euler figure eight emerges as a 'critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajectories. There are examples where the length of the curves along the same trajectory is not a monotone fiunction of the flow--time. It is shown how to determine the elliptic modulus of the critical curves in all the standard cases. The modulus p must satisfy 2E(p)/K(p) -1 ? gl/L when the space is limited to curves of fixed length L and the endpoints are separated by the vector g.