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Biholomorphic maps between Teichmüller spaces

Markovic, Vladimir (2003) Biholomorphic maps between Teichmüller spaces. Duke Mathematical Journal, 120 (2). pp. 405-431. ISSN 0012-7094. doi:10.1215/S0012-7094-03-12028-1.

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In this paper we study biholomorphic maps between Teichmüller 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 Teichmüller 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 Teichmüller space Teich(M) must be geometric. This result generalizes the previously known results (see [2], [5], [7]) 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.

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Additional Information:© 2003 Duke University Press. Received 17 May 2002. Revision received 17 January 2003. Author’s research partly supported by an Engineering and Physical Sciences Research Council Advanced Research Fellowship. I would like to thank the referee for a number of valuable comments and suggestions. In particular, I am grateful to the referee for pointing out to me that Lemma 3.2 holds for all Riemann surfaces of nonexceptional type. Also, the previous version of this paper contained the proof of Theorem 2.1. I am thankful to the referee for bringing to my attention the paper of Rudin [8], where this theorem was originally proved.
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Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Issue or Number:2
Classification Code:2000 Mathematics Subject Classification: Primary 30F60; Secondary 30F20
Record Number:CaltechAUTHORS:20170508-095311372
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Official Citation:Markovic, Vladimir. Biholomorphic maps between Teichmüller spaces. Duke Math. J. 120 (2003), no. 2, 405--431. doi:10.1215/S0012-7094-03-12028-1.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77253
Deposited By: Tony Diaz
Deposited On:08 May 2017 19:55
Last Modified:15 Nov 2021 17:29

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