The PI will continue the work of generalizing classical geometric variational problems in Riemannian geometry to include singular geometry. The main focus is the harmonic map theory from singular domains to non-positively curved metric spaces. Harmonic maps are important as analytical objects which can be used to represent geometric, topological and algebraic objects associated to a space and have numerous actual and potential applications in geometry, topology, and algebra, as well as other fields in mathematics. In a joint work with G. Daskalopoulos, we investigate existence and regularity properties of harmonic maps from polyhedral domains. The main applications are in geometric superrigidity, geometric group theory, character varieties and Hodge theory. In a project with S. Yamada, we study minimal surfaces with prescribed singularity complimenting the existing theory in geometric measure theory. This work includes the singular version of the classical Plateau problem as well as the equivariant minimal surface problem.
The primary object of study in the proposed projects is a harmonic map. A map is a geometric representation of one space into another. There is a natural notion of energy of a map which measures the total stretch. A harmonic map achieves equilibrium position with respect to this energy. For example, a configuration of a rubber sheet spread out over a bumpy surface of a rock can be understood as a map. The energy is the total tension of the rubber and the configuration of least tension is a harmonic map. A rubber sheet, without any outside forces acting on it, will naturally arrange itself into a position given by a harmonic map. Thus, a harmonic map is a natural and canonical choice in a given set of maps. The understanding of the properties of this map is mathematically interesting in itself but also essential as it represents many important phenomena in the natural sciences.
