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

小行星9701303 这个项目涉及各种黎曼几何性质的Teichmuller空间，这反过来又给出了一个新的方法来参数化的模紧黎曼曲面。更具体地说，研究者将利用Teichmuller空间的几何特征来研究映射到曲面的纤维化，并利用单连通完全双曲曲面的模来识别圆的微分同胚群。其他应用包括欧氏三球面中的极小嵌入曲面。 黎曼曲面是曲面的抽象;紧致黎曼曲面是椒盐卷饼状曲面（具有任意数量的孔），其中角度是明确定义的。这些表面最初出现在解决两个变量的多项式方程。

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.