Harmonic Maps and Their Applications

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Most objects in nature are not smooth. Yet most existing literature in geometry deal with smooth geometric objects such as a plane, a sphere, a saddle surface and their generalizations. In this NSF funded project, the PI proposes to study singular geometry by analyzing maps into and between them that are optimal in the sense that they minimize energy. The analysis of these maps will help us understand symmetry properties that exists in certain singular spaces. The project will advance our scientific knowledge in the understanding of non-singular geometry; in particular, the PI's research will lead to a greater understanding of the natural world. The project also has an education component and supports diversity by teaching and advising graduate students, post-doc and early career mathematicians especially those that are underrepresented in the STEM fields. 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 parametrized 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. The celebrated work of Eells and Sampson launched an explosion of research in harmonic maps between Riemannian manifolds. Many important applications were found, particularly in minimal surface theory, Teichmuller theory and rigidity questions. A more recent development is the harmonic map theory for non-smooth spaces. The seminal works of Gromov-Schoen and Korevaar-Schoen study energy minimizing maps in the case when the target is an non-positively curved (NPC) space. These energy minimizing maps are referred to as harmonic maps. The PI will extend this harmonic map theory for singular geometry and to apply it to solve problems in other fields.