## Duke Mathematical Journal

- Duke Math. J.
- Volume 120, Number 2 (2003), 229-467

### Biholomorphic maps between Teichmüller spaces

#### Abstract

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*.

#### Article information

**Source**

Duke Math. J. Volume 120, Number 2 (2003), 405-431.

**Dates**

First available: 16 April 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.dmj/1082138590

**Digital Object Identifier**

doi:10.1215/S0012-7094-03-12028-1

**Zentralblatt MATH identifier**

02051864

**Mathematical Reviews number (MathSciNet)**

MR2019982

**Subjects**

Primary: 30F60: Teichmüller theory [See also 32G15]

Secondary: 30F20: Classification theory of Riemann surfaces

#### Citation

Markovic, Vladimir. Biholomorphic maps between Teichmüller spaces. Duke Mathematical Journal 120 (2003), no. 2, 405--431. doi:10.1215/S0012-7094-03-12028-1. http://projecteuclid.org/euclid.dmj/1082138590.