A Study of Moduli of Riemann Surfaces via the Weil-Petersson Geometry of Teichmuller Spaces

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9701303 Tian This project concerns various Riemannian geometric properties of the Teichmuller space, which in turn gives a new way to parametrize the moduli of compact Riemann surfaces. More specifically, the investigator is to use the geometric characteristics of the Teichmuller space to study fibrations of maps into surfaces, and to identify the diffeomorphism group of the circle with the moduli of simply connected complete hyperbolic surfaces. Other applications include minimal embedded surfaces in Euclidean three sphere. Riemann surfaces are an abstraction of curved surfaces; compact Riemann surfaces are pretzel-like surfaces (with any number of holes) where angles are well-defined. These surfaces originally arose in connection with solving polynomial equations in two variables.