Geometry and applications of deformations of Riemann surfaces

研究者将继续研究Teichmueller空间的Weil-Petersson几何的测地线（最短路径）、凸性和曲率（几何的基本描述）。研究者已经证明，Teichmueller空间是一个无限多面体，具有无限数量的顶点和测地线面以直角相交。给出了确定多面体全对称群的Masur-Wolf结果的一个简短证明。研究者提供了面部附近几何形状的详细信息。双曲曲面上的封闭测地线（最短封闭路径）的长度是在Teichmueller空间中定位点的参数。研究者将继续他的计划，利用长度变化的分析来描述Teichmueller空间的综合和微分几何。这种描述比研究者先前的方法更简单，目前被许多研究者使用。研究者正在用长度的变化来描述曲率。研究者还将研究调和（最小总平方能量）映射到无限多面体和从无限多面体。他还将继续研究Weil-Petersson测地动力系统，以及黎曼曲面模空间的相交理论。黎曼曲面是一个二维曲面，它在每个点上都有角度测量的概念。具有至少两个柄的黎曼曲面具有确定距离、最短路径和振动膜算子的非欧几里得（双曲）几何。黎曼曲面有各种各样的形状，由厚子区域和薄子区域的相对位置描述。Teichmueller空间是具有给定柄数的黎曼曲面的所有可能形状的空间。单个黎曼曲面的双曲几何导致了Teichmueller空间的Weil-Petersson几何。黎曼曲面为振动膜、粒子传播和分形提供了模型。

The investigator will continue his study of geodesics (shortest paths), convexity and curvature (the fundamental descriptor for a geometry) for the Weil-Petersson geometry for Teichmueller space. The investigator has already shown that Teichmueller space is an infinite polyhedron with an infinite number of vertices and geodesic faces meeting at right angles. The description gives a short proof of the Masur-Wolf result determining the full symmetry group of the polyhedron. The investigator has provided detailed information on the geometry near the faces. Lengths of closed geodesics (shortest closed paths) on a hyperbolic surface are parameters for locating a point in Teichmueller space. The investigator will continue his program of using analysis of variations of the lengths to describe the synthetic and differential geometry of Teichmueller space. The description is simpler than the investigator's prior approach, which is presently used by a number of researchers. The investigator is developing a description of curvature in terms of variations of lengths. The investigator will also study harmonic (least total square energy) mappings to and from the infinite polyhedron. He will also continue to study the Weil-Petersson geodesic dynamical system, as well as intersection theory for the moduli space of Riemann surfaces.A Riemann surface is a two-dimensional surface with a notion of angle measure at each point. A Riemann surface with at least two handles is endowed with non Euclidean (hyperbolic) geometry determining distance, shortest paths and a vibrating membrane operator. Riemann surfaces come in various shapes, described by the relative locations of thick and thin subregions. Teichmueller space is the space of all possible shapes for a Riemann surface with a given number of handles. The hyperbolic geometry of individual Riemann surfaces leads to the Weil-Petersson geometry for Teichmueller space. Riemann surfaces provide models for vibrating membranes, propagating particles and fractals.