The Teichmüller spaces are distinct

Patterson, David B.

doi:10.1090/S0002-9939-1972-0299774-5

The Teichmüller spaces are distinct
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Proc. Amer. Math. Soc. 35 (1972), 179-182 Request permission

Erratum: Proc. Amer. Math. Soc. 38 (1973), 668.

Abstract:

The Teichmüller space $T(g,n)$ of a compact Riemann surface of genus g with n punctures is a complex manifold of $\dim = 3g - 3 + n$. In any given dimension, there are a finite number of these Teichmüller spaces and it is natural to ask if all of these are distinct (up to biholomorphic equivalence). We have shown here that with the exception of two special cases in dimensions 1 and 3 all of these spaces are distinct, that is, $T(g,n)$ is not biholomorphically equivalent to $T(g’,n’)$ unless $g’ = g$ and $n’ = n$.

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