Harmonic maps approach to rigidity problems

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In this project, the PI will study geometric spaces that have singularities. Most of the existing literature in geometry deal with non-singular, thus smooth, geometric objects such as a plane, a sphere, a saddle surface and their generalizations. On the other hand, most objects in nature are non-smooth with singularities of varying degrees of complexity. The PI proposes to study certain singular geometric spaces by analyzing maps into and between them that are optimal in the sense that they minimize energy. From the analysis of these maps, the PI proposes to study symmetry properties of the singular geometric spaces. Such results are of interest in mathematics and physics and will lead to a greater understanding of the natural world.A natural notion of energy for a map between geometric spaces is defined by measuring the total stretch of the map at each point of the domain and then integrating. Harmonic maps are critical points of the energy functional. They can be seen as both a generalization of harmonic functions in complex analysis and a higher dimensional analogue of parameterized geodesics in Riemannian geometry. Next to totally geodesic maps, harmonic maps are perhaps the most natural way to map a given geometric space into another. In this proposal, the PI proposes to extend the harmonic map theory for singular geometry and apply it to solve problems in other fields. When studying harmonic maps, natural questions that arise are: Under what conditions are harmonic maps regular? If it is not regular, what is the Hausdorff dimension of the singular set? Is the singular set rectifiable? We consider these questions in the singular setting (i.e. the domain and/or target space of a harmonic map not necessarily a smooth Riemannian manifold). By gaining a better understanding of these issues, we realize the potential applications of harmonic maps in the singular setting that include rigidity problems in geometric group theory and understanding degeneration of hyperbolic structures.