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

本课题研究了Teichmuller空间的各种黎曼几何性质，从而为紧化黎曼曲面模的参数化提供了一种新的方法。更具体地说，研究者是利用Teichmuller空间的几何特征来研究映射成曲面的纤曲，并用单连通完全双曲曲面的模来识别圆的微分同构群。其他应用包括欧几里得三球面的最小嵌入表面。黎曼曲面是曲面的抽象；紧致黎曼曲面是类似椒盐卷饼的曲面（具有任意数量的孔），其角度是明确定义的。这些曲面最初是在求解两个变量的多项式方程时出现的。

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.