Harmonic maps into and between singlar spaces

Johns Hopkins University, Baltimore MD

Investigators

Abstract

Proposal DMS-0306212 PI: Chikako Mese (Connectitcut College) Title: Harmonic maps into and between singular spaces Abstract The principal investigator proposes to study harmonic maps into and between singular spaces. The classical theory of harmonic maps deals with maps between Riemannian manifolds. More recently, the importance of considering singular domains and targets has been discovered. Harmonic map theory in singular spaces was initiated by Gromov and Schoen and their analysis of harmonic maps into non-positively curved Riemannian simplicial complexes combined with Corlette's vanishing theorem is the basis of their proof of p-adic super-rigidity. The study of harmonic maps into singular targets was further generalized by Korevaar and Schoen and independently by Jost. Further generalization is to consider singular domains. In this project, we study harmonic maps from a simplicial polyhedron to a metric space of non-positive curvature. A fundamental question is the regularity of these maps, and our goal is to show that these maps are smooth enough to be useful in many applications. In particular, we hope to bring harmonic map theory and holomorphic quadratic differentials into the study of finitely generated groups. More precisely, we will use harmonic maps from a two-dimensional simplicial complex to understand finitely generated groups from their actions on R-trees. This point of view is important in the study of combinatorial group theory and three-dimensional topology. Harmonic maps will also be used to investigate compactifications of the Teichmuller space of a compact surface. Finally, the study of minimal surfaces will be considered as an extension of the generalized harmonic map theory. The proposed work contributes to the basic understanding of geometric variational problems. Mathematicians have devoted large effort in developing variational methods and the successes of these investigations have laid the foundations of many branches of sience. There is a natural notion of energy associated to maps between certain spaces and, in this project, we study its critical points which are called harmonic maps. They have shown to be extremely useful as an analytic tool in geometry. The generalization of harmonic maps between smooth spaces to non-smooth spaces promises to yield many more applications. We investigate the extent to which the classical methods in harmonic maps can be carried over to the singular setting. The applications of the generalized theory make these questions relevant to a broad mathematical community.