Harmonic maps into and between singlar spaces

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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.