Abstract
We study the class of holomorphic and isometric submersions between finite-type Teichmüller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map \({{\cal T}_{g,n}} \to {{\cal T}_{g,m}}\) obtained by filling in punctures. This generalizes a classical result of Royden and Earle—Kra asserting that biholomorphisms between finite-type Teichmüller spaces arise from mapping classes. As a key step in the argument, we prove that any ℂ-linear embedding Q(X) ↪ 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 Teichmüller spaces.
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Acknowledgements
The first author would like to thank Martin Möller for a helpful discussion. The second author is grateful to Lizhen Ji for raising the main questions and for helpful discussions. The second author is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE#1256260.
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Gekhtman, D., Greenfield, M. Isometric submersions of Teichmüller spaces are forgetful. Isr. J. Math. 247, 499–517 (2022). https://doi.org/10.1007/s11856-021-2276-0
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DOI: https://doi.org/10.1007/s11856-021-2276-0