The universal family of punctured Riemann surfaces is Stein

TL;DR

This paper proves that the universal family of n-punctured Riemann surfaces of genus g is a Stein manifold, exploring its complex function theory and embedding properties.

Contribution

It establishes the Stein property for the universal Teichmüller family and investigates its algebraic and holomorphic function spaces.

Findings

The space of fibrewise algebraic functions is dense in holomorphic functions.

Existence of fibrewise algebraic embeddings into Euclidean space.

A relative Oka principle for fibrewise algebraic maps.

Abstract

We show that the universal Teichm\"uller family of n-punctured compact Riemann surfaces of genus g is a Stein manifold for any n>0. We describe its basic function theoretic properties and pose several challenging questions. We show in particular that the space of fibrewise algebraic functions on the universal family is dense in the space of holomorphic functions, and there is a fibrewise algebraic map of the universal family in a Euclidean space which restricts to a proper embedding on every fibre. We also obtain a relative Oka principle for holomorphic fibrewise algebraic maps of the universal family to any flexible algebraic manifold.