Analytic Aspects of the Harmonic Map Problem
Analytic Aspects of the Harmonic Map Problem
复制标题
DOI：
10.1007/978-1-4612-1110-5_17
复制
发表时间：
1984
期刊：
影响因子：
--
通讯作者：
R. Schoen
中科院分区：
文献类型：
--
作者：
R. Schoen
微分几何中一个基本的非线性对象是流形之间的映射。如果流形有黎曼度量，那么很自然地为映射选择表示，这些映射尊重流形的度量结构。经验表明，应该选择作为变分积分的极小点或临界点的映射。在已提出的积分中，能量最引起分析家、几何学家和数学物理学家的兴趣。它的临界点，调和映射，是一些几何兴趣。他们也被证明是有用的应用微分几何。特别应该提到它们在经典极小曲面理论中所发挥的重要作用。其次，通过[S]，[SiY]中给出的调和映射在K~ hler几何中的应用，说明调和映射作为几何分析工具的有效性。在作者看来，有充分的理由对与这个问题有关的技术和结果在几何学未来发展中所起的作用感到乐观。本文既是一份调查报告，也是一份研究报告。这是一个调查，其中许多讨论的结果是相当古老和众所周知的。我们没有试图写一个完整的调查的主题，而是选择了主题，我们觉得可以统一或简化。特别是，我们给或多或少完整的证明，我们讨论的结果。在第一节中，我们用变分方法建立了调和映射问题，并讨论了弱调和映射和能量积分的不动点。在第二节中，我们得到了光滑调和映射的各种先验估计。我们证明
A fundamental nonlinear object in differential geometry is a map between manifolds. If the manifolds have Riemannian metrics, then it is natural to choose representaives for maps which respect the metric structures of the manifolds. Experience suggests that one should choose maps which are minima or critical points of variational integrals. Of the integrals which have been proposed, the energy has attracted most interest among analysts, geometers, and mathematical physicists. Its critical points, the harmonic maps, are of some geometric interest. They have also proved to be useful in applications to differential geometry. Particularly one should mention the important role they play in the classical minimal surface theory. Secondly, the applications to K~ hler geometry given in [S],[SiY] illustrate the usefulness of harmonic maps as analytic tools in geometry. It seems to the author that there is good reason to be optimistic about the role which the techniques and results related to this problem can play in future developments in geometry. This paper is both a survey and a research paper. It is a survey in that many of the results which are discussed are quite old and well known. We have not attempted to write a complete survey of the subject, but instead have chosen topics which we feel can be unified or simplified. In particular, we give more or less complete proofs of the results which we discuss. In Section 1 we formulate the harmonic map problem variationally, and discuss weakly harmonic maps and stationary points of the energy integral. In Section 2 we derive various a priori estimates on smooth harmonic maps. We prove