Mathematical Sciences: Analytical Geometry of Families of Riemann Surfaces

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Four topics in this continuing investigation into the geometry and potential theory of Riemann surfaces will be covered by this project. The first of these concerns compact Riemann surfaces, combining Riemannian geometry, conformal geometry and algebraic geometry. The interplay between these concepts is strongest when the surface is considered with the Arakelov metric attached. In terms of this metric a Green's function is determined. One object of this research is to study the behavior of the Green's function for a degenerating family of surfaces. Recently, a parametrization mapping of holomorphic quadratic differentials on a surface with a hyperbolic metric has been constructed with range in the genus g Teichmuller space. It is known to be real analytic. Work will now proceed to extend this parametrization to the case where the surface is replaced by a noded Riemann surface. The object is to examine further the degeneration of hyperbolic metrics as well as the degeneration of harmonic maps. Further work on Teichmuller space centers on finding a boundary for the space and a kernel function (of two variables) which transforms by the diagonal action of the mapping class group. In addition, it has been noted that the degeneration of the spectrum for hyperbolic Laplace-Beltrami operators leads to improved understanding of the degeneration of the hyperbolic metric. Although the spectrum of the limit is not the limit of the spectrum, the possibility of obtaining one from the other is to be investigated. Of special interest will be the formation of continuous spectrum and the analysis of discrete spectrum in the limit. Some of this work relates to earlier studies on string theory - further applications will be considered as the research progresses.