Harmonic maps into and between singlar spaces

提案DMS-0306212PI：Chikako Mese(连通学院)标题：到奇异空间和奇异空间之间的调和映射主要研究者建议研究到奇异空间和奇异空间之间的调和映射。经典的调和映射理论处理黎曼流形之间的映射。最近，人们发现了考虑奇异域和目标的重要性。奇异空间中的调和映射理论最早是由Gromov和Schoen提出的，他们将调和映射分析为非正曲黎曼单纯复形，并结合Corlette的消零定理，是他们证明p-进超刚性的基础。到奇异目标的调和映射的研究由Koreva和Schoen进一步推广，并由Jost独立完成。进一步的推广是考虑奇异域。在这个项目中，我们研究了从单纯多面体到非正曲率度量空间的调和映射。一个基本的问题是这些映射的正则性，我们的目标是证明这些映射足够光滑，在许多应用中都是有用的。特别是，我们希望将调和映射理论和全纯二次微分引入到有限生成群的研究中。更确切地说，我们将使用来自二维单纯复形的调和映射来理解有限生成的群在R-树上的作用。这一观点在组合群论和三维拓扑学的研究中是很重要的。调和映射也将被用来研究紧致曲面的TeichMuller空间的紧化。最后，极小曲面的研究将被视为广义调和映射理论的推广。所提出的工作有助于对几何变分问题的基本理解。数学家在发展变分方法方面付出了巨大的努力，这些研究的成功奠定了许多科学分支的基础。某些空间之间的映射有一个自然的能量概念，在这个项目中，我们研究它的临界点，称为调和映射。它们已被证明是几何学中的一种非常有用的分析工具。光滑空间到非光滑空间的调和映射的推广有望产生更多的应用。我们考察了调和映射中的经典方法在多大程度上可以推广到奇异情形。广义理论的应用使这些问题与广泛的数学社区相关。

Proposal DMS-0306212PI: Chikako Mese (Connectitcut College)Title: Harmonic maps into and between singular spacesAbstract The principal investigator proposes to study harmonic maps into andbetween singular spaces. The classical theory of harmonic maps deals withmaps between Riemannian manifolds. More recently, the importance ofconsidering singular domains and targets has been discovered. Harmonic maptheory in singular spaces was initiated by Gromov and Schoen and theiranalysis of harmonic maps into non-positively curved Riemannian simplicialcomplexes combined with Corlette's vanishing theorem is the basis of theirproof of p-adic super-rigidity. The study of harmonic maps into singulartargets was further generalized by Korevaar and Schoen and independently byJost. Further generalization is to consider singular domains. In thisproject, we study harmonic maps from a simplicial polyhedron to a metricspace of non-positive curvature. A fundamental question is the regularityof these maps, and our goal is to show that these maps are smooth enough tobe useful in many applications. In particular, we hope to bring harmonicmap theory and holomorphic quadratic differentials into the study offinitely generated groups. More precisely, we will use harmonic maps from atwo-dimensional simplicial complex to understand finitely generated groupsfrom their actions on R-trees. This point of view is important in thestudy of combinatorial group theory and three-dimensional topology.Harmonic maps will also be used to investigate compactifications of theTeichmuller space of a compact surface. Finally, the study of minimalsurfaces will be considered as an extension of the generalized harmonic maptheory. The proposed work contributes to the basic understanding of geometricvariational problems. Mathematicians have devoted large effort indeveloping variational methods and the successes of these investigationshave laid the foundations of many branches of sience. There is a naturalnotion of energy associated to maps between certain spaces and, in thisproject, we study its critical points which are called harmonic maps. Theyhave shown to be extremely useful as an analytic tool in geometry. Thegeneralization of harmonic maps between smooth spaces to non-smooth spacespromises to yield many more applications. We investigate the extent to whichthe classical methods in harmonic maps can be carried over to the singularsetting. The applications of the generalized theory make these questionsrelevant to a broad mathematical community.