Harmonic Maps, Minimal Surfaces, and Rigidity Problems

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The proposed research concerns the development of harmonic map theory in the setting when the domain and/or the target spaces are singular spaces. The motivation of such an extension of harmonic maps is for their application in superrigidity, geometric group theory, minimal surface theory and character varieties. The PI proposes to explore the interplay between analysis and group theory via harmonic maps. From the analysis point of view, it is a central problem to determine under what conditions harmonic maps are regular and, if not, to determine the structure of the singular sets. From a group theory point of view, harmonic map theory is an important tool in studying rigidity questions. Among the specific problem that will be investigated using harmonic maps are: rigidity properties of NPC (non-positively curved) spaces, quadratic differentials and Teichmuller theory of complexes and the theory of singular minimal surfaces. The notion of energy has its origins in the mathematical description of the physical world. Roughly speaking, the energy of a map between two spaces measures the total stretch of the map, and harmonic maps are the critical points of the energy functional. Thus, the understanding of harmonic maps is of great interest to both mathematicians and physicists. Harmonic maps are important because they are analytical objects that can be used to represent geometric, topological and algebraic objects associated to a space. They have numerous actual and potential applications in many fields of mathematics or well as in other sciences.