Introduction to Teichmüller theory, old and new, II

6 Athanase Papadopoulos
maps between infinite-dimensional Teichmüller spaces that are reported on here by
Fletcher and Markovic, since these are results that hold in both the finite- and in
the infinite-dimensional cases. Fletcher and Markovic study biholomorphic maps
between Teichmüller spaces by examining their induced actions on cotangent spaces
(and Bergman spaces). In the finite-dimensional case, the idea of studying the action
on cotangent space is already contained in the early work of Royden. The action of
a biholomorphic map induces a C-linear isometry between Bergman spaces. Fletcher
and Markovic report on a rigidity result, whose most general form is due to Markovic,
and with special cases previously obtained by Earle & Kra, Lakic and Matsuzaki.
The result says that any surjective C-linear isometry between the Bergman spaces
A 1 (M) and A 1 (N) of two surfaces M and N is geometric, except in the case of some
elementary surfaces. Roughly speaking, the word “geometric” means here that the
isometry is a composition of two naturally defined isometries between such spaces,
viz. multiplication by a complex number of norm one, and an isometry induced by
the action of a conformal map between the surfaces. A corollary of this result is that
the biholomorphic automorphism group of the Teichmüller space of a surface of nonexceptional
(finite or infinite) type can be naturally identified with the mapping class
group of that surface.
As in the finite-dimensional case, this result reduces the study of biholomorphic
homeomorphisms between Teichmüller spaces to the study of linear isometries between
some Banach spaces. In the course of proving this result, a proof is given of the
fact that the Kobayashi and the Teichmüller metrics on (finite- or infinite-dimensional)
Teichmüller space agree, again generalizing a result obtained by Royden and completed
by Earle & Kra for finite type Riemann surfaces.
Chapter 2 of this volume also contains the proof of a local rigidity result due
to Fletcher, saying that the Bergman spaces of any two surfaces whose Teichmüller
spaces are infinite-dimensional are always isomorphic, and that any two infinitedimensional
Teichmüller spaces are locally bi-Lipschitz equivalent. More precisely,
Fletcher proved that the Teichmüller metric on every Teichmüller space of an infinitetype
Riemann surface is locally bi-Lipschitz equivalent to the Banach space l ∞ of
bounded sequences with the supremum norm.
1.3 Holomorphic families
A holomorphic family of Riemann surfaces of type (g, n) is a triple (M,π,B)defined
as follows:
M is a 2-dimensional complex manifold (topologically, a 4-manifold);
B is a Riemann surface;
π : M → B is a holomorphic map;
for all t ∈ B, the fiber St = π −1 (t) is a Riemann surface of genus g with n
punctures;
the complex structure on St depends holomorphically on the parameter t.