Isometric submersions of Teichm\"uller spaces are forgetful

Dmitri Gekhtman; Mark Greenfield

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arXiv · 1901.02586

Isometric submersions of Teichm\"uller spaces are forgetful

Dmitri Gekhtman·Mark Greenfield

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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.

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BibTeX RIS

Dmitri Gekhtman, Mark Greenfield. 2019-01-09. Isometric submersions of Teichm\"uller spaces are forgetful. https://arxiv.org/abs/1901.02586

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