Harmonic maps into and between singlar spaces

提案DMS-0306212 PI：Chikako Mese（Connectitcut学院）标题：调和映射到奇异空间和奇异空间之间摘要主要研究者提出研究调和映射到奇异空间和奇异空间之间。 调和映射的经典理论研究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是由Gromov和Schoen开创的，他们对调和映射到非正曲黎曼单复形的分析结合Corlette的消失定理是他们证明p-adic超刚性的基础。 Korevaar和Schoen以及Jost独立地进一步推广了对奇异目标的调和映射的研究。 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，在许多应用中是有用的。 特别地，我们希望把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，我们将使用来自二维单纯复形的调和映射来理解R生成群在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很重要，调和映射也将被用来研究紧曲面的Teichmuller空间的紧化。 最后，极小曲面的研究将被认为是广义调和映射理论的推广。 本文的工作有助于对几何变分问题的基本认识。 数学家们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。有一个自然的概念，能量与某些空间之间的映射，在这个项目中，我们研究其临界点，这被称为调和映射。它们已经被证明是非常有用的几何分析工具。 光滑空间到非光滑空间的调和映射的推广有希望产生更多的应用。我们研究调和映射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界有关。

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.