Uniformization By Classical Schottky Groups

跨越超越鸿沟：从 Schottky 群到代数曲线

Crossing the transcendental divide: from Schottky groups to algebraic curves

本文构造亏格 $g\geq 3$ 且带对称性与 $M$-curve 实结构的非超椭圆 Schottky Riemann 面，并数值计算其对应代数曲线与 Riemann 矩阵，以沟通解析与代数表示。

On neoclassical Schottky groups

The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are "sufficiently complicated". We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus $3$ that cannot be unif…

A sufficiently complicated noded Schottky group of rank three

The theoretical existence of non-classical Schottky groups is due to Marden. Explicit examples of such kind of groups are only known in rank two, the first one by by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to obtain examples of non-classical Schottky groups in any rank. The method assumes the knowledge of some algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups, whose existence was stated. In this paper we provide an explicit construction of a sufficiently complicated noded Schottky grou…

经典 Schottky 群的收缩格扩张

Systolic lattice extensions of classical Schottky groups

本文构造了经典 Schottky 群的收缩格扩张，证明闭双曲流形收缩元实现的复平移长度具有稠密性，并给出构造非算术格的新方法。

McShane's identity for classical Schottky Groups

Greg McShane introduced a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. This was subsequently generalized by the authors to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical Schottky groups. As a consequence, we obtain some surprising new identities in the case of fuchsian Schottky groups. For classical Schottky groups of rank 2, we also give generalizations of the Weierstrass identities, first given by McShane.

Classical Two-parabolic T-Schottky groups

A $T$-Schottky group is a discrete group of Möbius transformations whose generators identify pairs of, possibly-tangent, Jordan curves on the complex sphere, ${\hat{\IC}}$. If the curves are Euclidean circles then the group is termed classical $T$-Schottky. We describe the boundary of the space of classical $T$-Schottky groups affording two parabolic generators within the larger parameter space of all $T$-Schottky groups with two parabolic generators. This boundary is surprisingly different from that of the larger space. It is analytic while the boundary of the larger space appears to be fract…

Kleinian groups of small Hausdorff dimension are classical Schottky groups I

In this paper we prove that there exists a positive number $λ>0$, such that any 2-generated Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups.

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

This paper investigates the combinatorial $α$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $α$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. To handle…

On smooth moduli space of Riemann surfaces

In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$ The main theorem of the paper is: Closed Riemann surfaces are uniformizable by Schottky groups of Hausdorff dimension less than one. This work seem to be the only paper in literature to study question of Riemann surface uniformization and its Hausdorff dimension. We develop new techniques of rational norm of homological marking of Riemann surface and, decomp…

A fake Schottky group in Mod(S)

We use the classical construction of Schottky groups in hyperbolic geometry to produce non-Schottky subgroups of the mapping class group.

度量曲面的均匀化：综述

Uniformization of metric surfaces: A survey

本文综述了度量曲面均匀化的最新进展，从经典共形均匀化到分形曲面上的拟对称与拟共形均匀化，最终介绍任意局部有限面积度量曲面的弱拟共形均匀化结果。

All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal positive number $λ>0$, such that any finitely-generated non-elementary Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups. We will generalize our previous technologies given in \cite{HS} to prove our general result. Our result can be consider as …

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterize…

均匀化作为Tannakian重构

Uniformization as Tannakian Reconstruction

通过构造内在的典范极大抛物型PSL2-Higgs对象，将双曲均匀化表述为Tannakian重构定理，并建立了抛物型非阿贝尔Hodge-Riemann-Hilbert对应，从而将Betti实现与均匀化共尾Fuchsian格表示等同。

高维Schottky空间的拓扑

The topology of Schottky spaces in higher dimensions

研究高维Schottky空间的拓扑结构，主要定理表明在边界维数（两倍秩）时，空间的基本群是二阶循环群的乘积，但整体单连通，并证明旋转对称核是强形变收缩核。

The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture

In this paper we provide the complete classification of convex cocompact Kleinian group of Hausdorff dimensions less than $1.$ In particular, we prove that every convex cocompact Kleinian group of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. The result implies that the converse of Burside's conjecture \cite{Burside} is true: All non-classical Schottky groups must have Hausdorff dimension $\ge1$. The prove of the theorem relies on the result of Hou \cite{Hou}.

The classification of Kleinian groups of Hausdorff dimensions at most one

In this paper we provide the complete classification of Kleinian groups of Hausdorff dimensions less than $1.$ In particular, we prove that every purely loxodromic Kleinian groups of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. As an application, the result of \cite{H} then implies that, every closed Riemann surface is uniformizable by a classical Schottky group. The prove relie on the result of Hou \cite{Hou}, and space of rectifiable $\G$-invariant closed curves.