Homotopy classes in Sobolev spaces and the existence of energy minimizing maps
Homotopy classes in Sobolev spaces and the existence of energy minimizing maps
复制标题
索博列夫空间中的同伦类和能量最小化映射的存在性
DOI：
10.1007/bf02392271
复制
发表时间：
1988
期刊：
影响因子：
3.7
通讯作者：
B. White
中科院分区：
文献类型：
--
作者：
B. White
考虑寻找像fM IDfl p这样的能量泛函（其中M和N是紧致黎曼流形和p~>l）的平稳映射f: M—>N的问题。这样的映射可以通过最小化泛函来找到，但如果我们在从M到N的所有映射中最小化，则最小值为0，并且只能通过常数映射获得。因此，为了找到非平凡的平稳映射，我们希望使用M和N的拓扑来定义从M到N的映射类，在这些映射类中我们可以最小化泛函。例如，可以尝试在给定同伦类中的映射之间最小化，但这通常是不可能的（除非p>dimM），因为一个同伦类中的映射的最小化序列可以收敛（在适当的弱拓扑中）到另一个同伦类中的映射。然而，在本文中，我们证明了映射之间的最小化是可能的，这些映射的限制到一个低维骨架的（三角形）M属于一个给定的同伦类。为了精确地说明结果，我们需要参考某些Sobolev空间和范数。假设M和N分别是欧几里得空间R M和R ~的子流形，令Lip （M， N）表示从M到N的lipschitz映射空间，并定义LI'p（M, R '）为所有函数fE LP（M, R N）的空间，使得存在函数
Consider the problem of finding maps f: M--->N that are stationary for an energy functional such as fM IDfl p (where M and N are compact riemannian manifolds and p~>l). Such maps may be found by minimizing the functional, but if we minimize among all maps from M to N, then the minimum is 0 and is attained only by constant maps. Thus in order to find nontrivial stationary maps, we would like to use the topology of M and N to define classes of maps from M to N in which we can minimize the functional. For instance one could try to minimize among maps in a given homotopy class, but this is not possible in general (unless p>dimM), since a minimizing sequence of mappings in one homotopy class can converge (in the appropriate weak topology) to a map in another homotopy class. However, in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of (a triangulation of) M belong to a given homotopy class. To state the results precisely we need to refer to certain Sobolev spaces and norms. We will assume without loss of generality that M and N are submanifolds of euclidean spaces R m and R ~, respectively, and we let Lip (M, N) denote the space of lipschitz maps from M to N. We define LI'p(M, R") to be the space of all functions fE LP(M, R n) such that there exist functions