Biholomorphic maps between Teichmuller spaces
UNSPECIFIED. (2003) Biholomorphic maps between Teichmuller spaces. DUKE MATHEMATICAL JOURNAL, 120 (2). pp. 405-431. ISSN 0012-7094Full text not available from this repository.
In this paper we study biholomorphic maps between Teichmuller spaces and the induced linear isometries between the corresponding tangent spaces. The first main result in this paper is the following classification theorem. If M and N are two Riemann surfaces that are not of exceptional type, and if there exists a biholomorphic map between the corresponding Teichmuller spaces Teich(M) and Teich(N), then M and N are quasiconformally related. Also, every such biholomorphic map is geometric. In particular, we have that every automorphism of the Teichmuller space Teich(M) must be geometric. This result generalizes the previously known results (see , , ) and enables us to prove the well-known conjecture that states that the group of automorphisms of Teich(M) is isomorphic to the mapping class group of M whenever the surface M is not of exceptional type. In order to prove the above results, we develop a method for studying linear isometries between L-1-type spaces. Our focus is on studying linear isometries between Banach spaces of integrable holomorphic quadratic differentials, which are supported on Riemann surfaces. Our main result in this direction (Theorem 1.1) states that if M and N are Riemann surfaces of nonexceptional type, then every linear isometry between A(1)(M) and A(1)(N) is geometric. That is, every such isometry is induced by a conformal map between M and N.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Journal or Publication Title:||DUKE MATHEMATICAL JOURNAL|
|Publisher:||DUKE UNIV PRESS|
|Official Date:||1 November 2003|
|Number of Pages:||27|
|Page Range:||pp. 405-431|
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