Isometric submersions of Teichm\"uller spaces are forgetful

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.