Alexandrov 空间之间的调和映射正则性理论及其应用

Alexandrov 空间是允许存在奇异集的几何对象，如凸多面体，黎曼 Orbifolds 等。另一方面，调和映射是几何分析的一个重要方向，在几何与代数方面都有重要应用。 1992 年, Abel 奖和 Wolf 奖获得者 M. Gromov 和美国科学院院士 R.Scheon 为了研究秩1半单李群 p-adic 超刚性（superrigidity）问题，发展了从光滑流形到非正曲率 Alexandrov 空间的调和映射理论。 1997 年，林芳华和德国科学院院士 J.Jost 独立开展了Alexandrov 空间之间的调和映射理论。. 在本项目中，我们继续深入 Alexandrov 空间之间的调和映射理论研究。 主要包含如下三个问题： （1）研究此类调和映射 Lipschitz 正则性；（2） 建立此类调和映射的 Bochner 公式；（3）应用于一些李群的超刚性问题。

Alexandrov spaces are a class of geometric objects which allow the existence of singularities, for example, convex polyhedrons and Riemannian orbifolds. On the other hand, the theory of harmonic maps, as one of important aspects of geometric analysis, has many applications in geometry and algebra. In 1992, to study the p-adic superrigidity of semi-simple Lie group of rank 1, M. Gromov, a Abel prize and Wolf prize winner, and R. Schoen developed a theory of harmonic maps from smooth manifolds to Alexandrov spaces with non-positive curvature. In 1997, Fanghua Lin and J. Jost considered this theory between Alexandrov spaces, independently.. In this project, we shall study intensively the theory of harmonic maps between Alexandrov spaces. Three problems are included: (1)We will study the Lipschitz regularity of these harmonic maps; (2) We want to establish a Bochner formula for these harmonic maps; (3) We will study some superrigidity of Lie groups as some applications of this theory.