Harmonic Maps into Spaces with an Upper Curvature Bound

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The everyday definition of the word “map” is “a diagrammatic representation of an area of land or sea showing physical features, cities, roads, etc.” Cartographers construct maps to reveal interesting spacial information about a geographical region. In a similar way, mathematicians construct maps between geometric spaces to discover interesting features of those spaces. Here, by a geometric space, we mean a space equipped with notions of angles, distances, areas, etc. They include Euclidean spaces which are often used to model our everyday physical world or non-Euclidean or Riemannian spaces which can be used as a large scale model of our universe. This award provides funding to study special maps between geometric spaces called harmonic maps that minimize a certain notion of energy. By analyzing harmonic maps, the PI aims to uncover properties of important geometric spaces that would lead to a greater understanding of the natural world. The mathematical theory of harmonic maps has been applied in diverse fields such as medicine (for example, in medical imaging) and computer science (for example, in computer vision), and has further potential applications aiding in the scientific progress and welfare of our society. The project also has an educational component and supports diversity by teaching and advising graduate students, post-doc and early career mathematicians especially those who are underrepresented in the STEM fields. The project focuses on harmonic maps in spaces with an upper curvature bound. A harmonic map between Riemannian manifolds is a solution to a certain system of elliptic partial differential equations (harmonic map equations) and is also a solution to a variational problem involving the Dirichlet energy. Since the emergence of modern geometric analysis as a core mathematical discipline, the harmonic map theory has been at the forefront of the field and important applications continue to be found. A more recent development is the study of harmonic maps into complete metric spaces satisfying an upper curvature bound. The goal of this project is to develop this theory in order to apply it to solve rigidity problems, to understand the structure of surface fibration over Kahler manifolds and projective varieties, and to study quasiconformal maps.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.