On smooth moduli space of Riemann surfaces

TL;DR

This paper investigates the smooth moduli space of closed Riemann surfaces, showing they can be uniformized by Schottky groups with Hausdorff dimension less than one, and introduces new techniques for this analysis.

Contribution

It demonstrates that all closed Riemann surfaces are uniformizable by Schottky groups of Hausdorff dimension under one, using novel methods involving rational norms and measure decomposition.

Findings

Closed Riemann surfaces are uniformizable by Schottky groups with Hausdorff dimension less than one.

Develops new techniques for analyzing the uniformization and Hausdorff dimension of Riemann surfaces.

Establishes the existence of period matrices in the smooth moduli space coordinates.

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 $\mathscr{M}_g$ 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, 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.