Universal moduli spaces of Riemann surfaces

TL;DR

This paper constructs a universal moduli space for all finite genus Riemann surfaces, equipped with metrics and measures, with applications in minimal surface theory and string theory, using advanced geometric and algebraic tools.

Contribution

It introduces a universal moduli space for Riemann surfaces of all finite genus, utilizing the Torelli map and compactifications, extending the classical moduli space framework.

Findings

Constructed a stratified moduli space for all finite genus Riemann surfaces.

Defined metrics and measures inducing Riemannian structures on each stratum.

Outlined applications to minimal surfaces and Bosonic string theory.

Abstract

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional locally symmetric space which is of interest in its own right. The key to our construction of the universal moduli space then is the Torelli map that assigns to every Riemann surface its Jacobian and its extension to the Satake-Baily-Borel compactifications.