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 @math 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 @math -linear embedding @math 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.
We mention also a result of Antonakoudis-Aramayona-Souto @cite_9 stating that any holomorphic map @math between moduli spaces is forgetful, as long as @math and @math . One can see this is as a parallel of our result, with our metric constraint replaced by an equivariance condition.
@cite_9
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We prove that every non-constant holomorphic mapeM& em&⊂g,p& sub&→ eM& em&⊂& g',p'& sub& between moduli spaces of Riemann surfaces is a forgetful map, provided that g ≥ 6 and g' ≤ 2g-2.