Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Earle, Clifford; Gardiner, Frederick

doi:10.1090/S0002-9947-96-01490-0

Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Authors: Clifford J. Earle and Frederick P. Gardiner
Journal: Trans. Amer. Math. Soc. 348 (1996), 1163-1190
MSC (1991): Primary 32G15; Secondary 30C62, 30C75
DOI: https://doi.org/10.1090/S0002-9947-96-01490-0
MathSciNet review: 1322950
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Abstract: Let $X$ and $Y$ be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by a quasiconformal homeomorphism of $X$ onto $Y$. These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.

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