Curve-straightening in closed Euclidean submanifolds
Curve-straightening in closed Euclidean submanifolds
复制标题
闭欧几里得子流形中的曲线拉直
DOI：
10.1007/bf02099668
复制
发表时间：
1991
影响因子：
2.4
通讯作者：
Anders Linnér
中科院分区：
文献类型：
--
作者：
Anders Linnér
与泛函总平方(测地线)曲率∫k2d有关的梯度向量场的负方向的流称为曲线拉直流。本文将考虑闭欧几里德子流形中的闭曲线空间。它将这些曲线空间定义为代表所有曲线的某些希尔伯特流形的子流形。主要结果是证明了定义在整个Hilbert流形上的一组特殊泛函的存在性，它具有以下四个性质：1.这些泛函的方向导数可以通过求解一个常微分方程组的初值问题来计算。2.通过为所使用的Soblev空间引入适当的Hilbert空间基，可以有效地计算梯度(但当然不是显式计算，除非在非常特殊的情况下)。3.梯度横跨与闭合曲线空间的切线空间垂直的空间。4.尽管这些梯度一般不是明确给出的，但仍然可以计算到与闭合曲线空间相切的空间上的投影。特别地，我们对∫k2ds的梯度做了这项工作。当所有细节都解决后，这就给了我们一个在闭合曲线空间中寻找临界点的算法(我们提供了这个算法)。目前尚不清楚这些轨迹是否实际上总是收敛到临界点。如果将泛函修改为包括长度的倍数，则泛函变为∫k2+λds，则对于λ>；0，上述收敛是已知的。曲线拉直流的激励性应用是使用它寻找(非平凡的)闭合(周期)测地线的可能性。请注意，如果λ=0，则闭合测地线是全局最小值。对于任何λ来说，测地线都是关键的，但还有其他关键点，即所谓的弹性曲线。给出了∫K2+λdsalong闭测地线的第二个变分公式。证明了与二阶导数相关的二次泛函是正定的，即使对于非零的λ，也是沿着某些特定流形中的某些闭测地线。
The flow in the negative direction of the gradient vector field associated with the functional total squared (geodesic) curvature ∫k2dsis the so-called curvestraightening flow. This paper will consider spaces of closed curves in closed Euclidean submanifolds. It will define these spaces of curves as submanifolds of certain Hilbert manifolds representing all curves. The main result will then be to show the existence of a particular set of functionals defined on the entire Hilbert manifold which have the following four properties: 1. The directional derivatives of these functionals may be computed by solving an initial value problem for a system of ordinary differential equations. 2. By introducing a suitable Hilbert space basis for the Sobolev spaces used, the gradients may be effectively computed (but of course not explicitly computed, except in very special cases). 3. The gradients span the space normal to the tangent space of the space of closed curves. 4. Despite the fact that these gradients in general are not given explicitly it is nevertheless possible to compute the projection onto the tangent space to the space of closed curves. In particular we do this for the gradient of ∫k2ds. When all details are worked out this gives us an algorithm (which we supply) for finding critical points in the space of closed curves. It is not known if the trajectories actually always converge to critical points. If the functional is modified to include a multiple of the length so the functional becomes ∫k2+λdsthen the above convergence is known for λ>0. The motivating application for the curvestraightening flow is the possibility of using it to find (non-trivial) closed (periodic) geodesics. Note that if λ=0 then a closed geodesic is a global minimum. For any λ, geodesics are critical but there are also other critical points, the so-called elastic curves. The paper concludes by deriving the second variation formula for ∫k2+λdsalong closed geodesics. The quadratic functional associated with the second derivative is shown to be positive definite even for non-zero λ along some closed geodesics in some particular manifolds of interest.