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

Isometric submersions of Teichm\"uller spaces are forgetful