Mathematical Sciences: Analytical Geometry of Families of Riemann Surfaces

在这个持续调查的四个主题 几何和潜在的理论黎曼曲面将涵盖 通过这个项目。 第一个是关于紧致黎曼的 曲面，结合黎曼几何，保形几何和 代数几何 这些概念之间的相互作用是 最强时，表面被认为是与阿拉克洛夫度量 附 根据该度量，绿色函数为 测定 本研究的一个目的是研究 退化曲面族的绿色函数。 最近，全纯二次型的参数化映射 在一个曲面上的微分与双曲度量一直是 在亏格g Teichmuller空间中的值域构造。 是 已知是真实的解析的。 现在将着手扩大这项工作， 参数化的情况下，表面被替换为 节点黎曼曲面 目的是进一步研究 双曲度量的退化以及 调和映射 Teichmuller空间的进一步工作集中在寻找一个 空间的边界和一个核函数（两个变量） 它通过映射类的对角作用进行变换 组 此外，人们注意到， 双曲Laplace-Beltrami算子谱导致 改进了对双曲型退化的理解 公制 虽然极限的频谱不是 光谱，从另一个获得一个的可能性是 接受调查 特别令人感兴趣的将是形成 连续谱和离散谱的分析 极限 其中一些工作与早期对弦的研究有关 理论-进一步的应用将被视为研究 进步。

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.