Mathematical Sciences: Applications of Variational Problems on Riemann Surfaces to Low-Dimensional Geometry

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The Principal Investigator proposes to study some variational problems on Riemann Surfaces that have applications to the study of the Moduli space of curves and three dimensional hyperbolic geometry. Specifically, he proposes to study an extremal length problem on a Riemann surface the solution of which would lead to new invariants of Riemann surfaces, and then new local functions on Moduli space. He also proposes to study the shapes of images of harmonic maps of surfaces in hyperbolic three-manifolds, including the differential-topological singularities of the images, and the dependence of the motion of the image surface in the three-manifold upon the motion of the domain in Teichmuller space.