Harmonic maps into and between singlar spaces

提案 DMS-0306212PI：Chikako Mese（康涅狄格学院）标题：奇异空间中和奇异空间之间的调和映射 摘要 首席研究员建议研究奇异空间中和奇异空间之间的调和映射。 调和映射的经典理论处理黎曼流形之间的映射。 最近，人们发现了考虑单一领域和目标的重要性。 奇异空间中的调和映射理论由 Gromov 和 Schoen 发起，他们对非正曲黎曼单纯复形的调和映射分析与 Corlette 消失定理相结合，是他们证明 p 进超刚性的基础。 Korevaar 和 Schoen 进一步推广了调和映射到奇异目标的研究，并由 Jost 独立推广。 进一步概括是考虑奇异域。 在这个项目中，我们研究从单纯多面体到非正曲率度量空间的调和映射。 一个基本问题是这些图的规律性，我们的目标是证明这些图足够平滑，可以在许多应用中使用。 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。 更准确地说，我们将使用二维单纯复形的调和映射来理解根据其在 R 树上的作用而生成的有限群。 这个观点对于组合群论和三维拓扑的研究很重要。调和图也将用于研究紧致曲面的Teichmuller空间的紧致化。 最后，最小曲面的研究将被视为广义调和映射理论的延伸。 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投入了大量的精力，这些研究的成功为许多科学分支奠定了基础。有一个与某些空间之间的映射相关的自然能量概念，在这个项目中，我们研究了它的临界点，称为调和映射。它们已被证明作为几何分析工具非常有用。 平滑空间到非平滑空间之间的调和映射的推广有望产生更多的应用。我们研究调和映射中的经典方法可以在多大程度上延续到奇异设置。 广义理论的应用使这些问题与广泛的数学界相关。

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.