Infinite Dimensional Morse Theory
Infinite Dimensional Morse Theory
复制标题
DOI：
10.1007/978-1-4612-0385-8_1
复制
发表时间：
1993
期刊：
影响因子：
--
通讯作者：
Kung-Ching Chang
中科院分区：
文献类型：
--
作者：
Kung-Ching Chang
莫尔斯理论的基本结果是莫尔斯不等式和莫尔斯柄体定理。在第4节中，它们建立在Banach Finsler流形或Hilbert黎曼流形上。本研究中的工具是变形定理，这将在第3节中介绍。第一节和第二节分别介绍了代数拓扑和无穷维流形的一些基本概念。熟悉背景材料的读者可以跳过这两个部分。孤立临界点上的Gromoll-Meyer理论在莫尔斯理论的应用中起着重要的作用，因为柄体定理中的非简并假设在具体问题中可能不成立。第五节系统地介绍Gromoll-Meyer理论，并考察分裂引理、同伦不变性定理、移位定理和Marino Prodi逼近定理。本章其余部分包括扩展的基本结果莫尔斯理论在不同的方向：在第6.1节，延伸到流形的边界以及职能满足一定的边界值条件，在第6.2节，延伸从流形的局部凸闭子集;并在第7节，职能与对称下紧李群行动。
The basic results in Morse theory are the Morse inequalities and the Morse handle body theorem. They are established on the Banach Finsler manifolds or on the Hilbert Riemannian manifolds in Section 4. The tool in this study is the deformation theorem, which is introduced in Section 3. Some preliminaries on algebraic topology and on infinite dimensional manifolds are reviewed in Sections 1 and 2 respectively. Readers who are familiar with the background material may skip over these two sections. Gromoll-Meyer theory on isolated critical points plays an important role in the applications of Morse theory because the nondegeneracy assumption in the handle body theorem might not hold for concrete problems. Section 5 is devoted to introducing Gromoll-Meyer theory systematically and examines the splitting lemma, the homotopy invariance theorem, the shifting theorem, and the Marino Prodi approximation theorem. The rest of the chapter consists of the extensions of the basic results of Morse theory in different directions: in Section 6.1, to the extension to manifolds with boundaries as well as to the functions satisfying certain boundary value conditions, in Section 6.2, to the extension from manifolds to the locally convex closed subsets; and, in Section 7, to functions with symmetry under a compact Lie group action.