# Isometric submersions of Teichm\"uller spaces are forgetful

## TL;DR

This paper characterizes holomorphic isometric submersions between finite-type Teichmüller spaces, showing they are mostly forgetful maps that fill in punctures, extending classical results about biholomorphisms.

## Contribution

It proves that such submersions are essentially forgetful maps, generalizing known results and introducing new techniques for analyzing quadratic differentials and Teichmüller space embeddings.

## Key findings

- Most holomorphic isometric submersions are forgetful maps filling punctures
- Any complex-linear embedding of quadratic differentials is a pull-back by a holomorphic map
- Method adapts techniques from infinite-type Teichmüller space studies

## Abstract

We study the class of holomorphic and isometric submersions between finite-type Teichm\"uller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map $\mathcal{T}_{g,n} \rightarrow \mathcal{T}_{g,m}$ obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichm\"uller spaces arise from mapping classes. As a key step in the argument, we prove that any $\mathbb{C}$-linear embedding $Q(X)\hookrightarrow Q(Y)$ between spaces of integrable quadratic differentials is, up to scale, pull-back by a holomorphic map. We accomplish this step by adapting methods developed by Markovic to study isometries of infinite-type Teichm\"uller spaces.