On smooth moduli space of Riemann surfaces

2016/10/10 by Yong Hou, Hou, Yong
Mathematics · #53 #57 #Advanced Algebra and Geometry #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.1610.03132

openalex publication_date 2016/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \mathscrM_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, decomposition of probability measures to prove our result. As an application of our theorem we have existence of period matrix of Riemann surface in coordinates of smooth moduli space.

Citations

• Tameness of hyperbolic 3-manifolds

Related

• Towards a proof of the Classical Schottky Uniformization Conjecture
• A tour through Mirzakhani's work on moduli spaces of Riemann surfaces
• A measure on the moduli space of super Riemann surfaces with Ramond punctures
• Moduli spaces and period mappings of genus one fibered K3 surfaces
• The Moduli space of Riemann Surfaces of Large Genus