Geometry in Teichmüller and moduli spaces

黎曼曲面在整个数学中无处不在。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连续的自然域出现，在几何中作为离散等距线群的双曲平面的商出现。虽然单个黎曼曲面可能是非常美丽的(例如克莱因四次曲面)，但TeichMuller理论的目的是在族中研究它们。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，而是一个轮流形(想想带角的流形)。为了简化问题，可以跟踪其他拓扑数据。因此，TeichMuller空间中的一个点是黎曼曲面，它与S的同伦类同态，称为标记。这个空间比模空间更简单：它同胚于6g-6维的欧几里得空间，其中g是S的亏格。Teichmuller空间中两点之间的Teichmuller距离度量了两个曲面距离共形等价的距离，即从一个曲面到另一个曲面需要变形多少角度。S保定向同胚同伦类的映射类群通过标号的变化作用于TeichMuller空间。由于此操作保持Teichmuller距离，因此后者下降到商上的度量，即模空间。*换言之，给定拓扑型的所有黎曼曲面的空间都有它自己的形状和几何。这种几何已被Ahlfors、Bers、Royden、Mumford、瑟斯顿、Masur、Minsky、McMullen、Mirzakhani和许多其他人广泛研究，并应用于Kleian群、复杂动力学和拓扑学。本研究计划的目标是进一步了解该几何学，并对其进行计算和可视化。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.