Geometry in Teichmüller and moduli spaces

黎曼曲面在数学中无处不在。它们在代数中作为多项式方程的解出现，在复分析中作为解析延拓的自然域出现，在几何中作为双曲平面的等距离散群的导数出现。虽然一个黎曼曲面可以非常漂亮（例如克莱因的四次曲面），但Teichmuller理论的目的是在家庭中研究它们。闭曲面S的模空间是与S同胚直到共形等价的黎曼曲面的集合。由于某些曲面比其他曲面具有更多的对称性，因此该空间具有复杂的结构：它不是流形，而是轨道（想想有角的流形）。为了简化问题，可以跟踪附加的拓扑数据。因此，Teichmuller空间中的一个点是一个黎曼曲面和一个与S同胚的同伦类，称为标记。这个空间比模空间简单：它与6 g- 6维的欧几里得空间同胚，其中g是S的亏格。Teichmuller空间中两点之间的Teichmuller距离测量了两个表面从共形等价到多远，即从一个表面到另一个表面需要扭曲多少角度。S的保向同胚的同伦类的映射类群通过变换标记作用在Teichmuller空间上。由于这个作用保持了Teichmuller距离，后者下降到商上的度量，这是模空间。** 换句话说，给定拓扑类型的所有黎曼曲面的空间都有自己的形状和几何。这一几何已被广泛研究的Ahlfors，Bers，罗伊登，芒福德，瑟斯顿，马苏尔，明斯基，McMullen，Mirzakhani和许多其他人，与应用克莱因集团，复杂的动力学和拓扑结构。本研究的目标是进一步理解这种几何，并将其计算和可视化，其中一个具体的目标是在Teichmuller空间中找到足够的全测地子流形来构造具有非空内部的紧致凸集。

Riemann surfaces are ubiquitous throughout mathematics. They appear in algebra as solutions to polynomial equations, in complex analysis as natural domains for analytic continuation, and in geometry as quotients of the hyperbolic plane by discrete groups of isometries. While a single Riemann surface can be quite beautiful (e.g. Klein's quartic), the purpose of Teichmuller theory is to study them in families.******The moduli space of a closed surface S is the set of Riemann surfaces homeomorphic to S up to conformal equivalence. Since some surfaces have more symmetries than others, that space has a complicated structure: it is not a manifold but an orbifold (think manifold with corners). To simplify matters, one can keep track of additional topological data. A point in Teichmuller space is thus a Riemann surface together with a homotopy class of homeomorphism to S, called a marking. This space is simpler than moduli space: it is homeomorphic to Euclidean space of dimension 6g - 6, where g is the genus of S. The Teichmuller distance between two points in Teichmuller space mesures how far the two surfaces are from being conformally equivalent, in terms of how much angles need to be distorted to go from one surface to the other. The mapping class group of homotopy classes of orientation-preserving homeomorphisms of S acts on Teichmuller space by change of marking. Since this action preserves Teichmuller distance, the latter descends to a metric on the quotient, which is moduli space. ******In other words, the space of all Riemann surfaces of a given topological type has a shape and a geometry of its own. That geometry has been studied extensively by Ahlfors, Bers, Royden, Mumford, Thurston, Masur, Minsky, McMullen, Mirzakhani and many others, with applications to Kleinian groups, complex dynamics and topology. The goal of the present research program is to further understand that geometry, and to compute and visualize it. One specific objective is to find enough totally geodesic submanifolds in Teichmuller space to construct compact convex sets with non-empty interior.