Isometric submersions of Teichmüller spaces are forgetful (original) (raw)

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 Tg,n → Tg,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.