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.
S. Antonakoudis @cite_14 was the first to study isometric submersions in the context of Teichm "uller theory. He proved that there is no holomorphic and Kobayshi-isometric submersion between a finite-dimensional Teichm "uller space and a bounded symmetric domain, provided each is of complex dimension at least two.
@cite_14
2145814138
We study isometric maps between Teichmuller spaces and bounded symmetric domains in their Kobayashi metric. We prove that every totally geodesic isometry from a disk to Teichmuller space is either holomorphic or anti-holomorphic; in particular, it is a Teichmuller disk. However, we prove that in dimensions two or more there are no holomorphic isometric immersions between Teichmuller spaces and bounded symmetric domains and also prove a similar result for isometric submersions.