# Finiteness theorems for holomorphic mappings from products of hyperbolic Riemann surfaces

## TL;DR

This paper proves that the set of non-constant holomorphic maps from products of hyperbolic Riemann surfaces to certain hyperbolic manifolds is finite, highlighting a finiteness property in complex geometry.

## Contribution

It establishes finiteness theorems for holomorphic mappings from products of hyperbolic Riemann surfaces into hyperbolic manifolds with bounded universal cover.

## Key findings

- The space of such holomorphic mappings is finite.
- Finiteness holds for dominant and non-constant mappings.
- Results apply to hyperbolic Riemann surfaces of finite type.

## Abstract

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds with universal cover a bounded domain is a finite set.