Geometry of Riemann Surfaces and Teichmüller Spaces

269 Pages·1992·13.032 MB·English

by  Mika Seppälä and Tuomas Sorvali (Eds.)

GEOMETRY OF RIEMANN SURFACES
AND TEICHMULLER SPACES
NORTH-HOLLAND MATHEMATICS STUDIES 169
(Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro, Brazil
and
University of Rochester
New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM LONDON NEW YORK TOKYO
GEOMETRY OF RIEMANN SURFACES
AND TEICHMULLER SPACES
Mika SEPPALA
Academy of Finland
Helsinki, Fin land
Tuomas SORVALI
University of Joensuu
Joensuu, Finland
1992
NORTH-HOLLAND -AMSTERDAM LONDON NEW YORK TOKYO
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Library of Congress Cataloging-in-Publication Data
Seppala. Mika.
Geometry of Riemann surfaces and Teichmuller spaces / Mika
Seppala, Tuomas Sorvali.
p. cm. -- (North-Holland Mathernalics studies , 169)
Incluoes bibliographical references and index.
ISBN 0-444-88846-2
1. Riem. ann surfaces. 2. Teichnuller spaces. I. Sorvali. Tuonas.
1944- 11. Title. 111. Series.
OA333. S42 1992
515(cid:146). 223--dc20 9 1-34760
CIP
ISBN: 0 444 88846 2
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Preface
This monograph grew out of a series of lectures held by the first author at
the University of Regensburg in 1986 and in 1987 and by the second author
at the University of Joensuu in 1990. This book would presumably not have
been written without the initiative of Professor Leopoldo Nachbin.
A large part of the present work has been carried out at the University
of Regensburg and at the Mittag-Leffler Institute. We thank these both
institutes for their warm hospitality.
Finally we thank Ari Lehtonen for several figures, especially for his in-
triguing illustration of the Klein bottle.
In Helsinki and in Joensuu, Finland
August 1991
Mika Seppala Tuoinas Sorvali
1
This Page Intentionally Left Blank
Introduction
The moduli problem is to describe the structure of the space of isomorphism
classes of Riemann surfaces of a given topological type. This space is known
as the moduli space. It has been in the center of pure mathematics for
more than 100 years now. In spite of its age, this field still attracts lots of
attention. The reason lies in the fact that smooth compact Riemann surfaces
are simply complex projective algebraic curves. Therefore the moduli space
of compact Riemann surfaces is also the moduli space of complex algebraic
curves. This space lies in the intersection of many fields of mathematics
and can, therefore, be studied from many different points of view.
Our aim is to get information about the structure of the moduli space
using as concrete and as elementary methods as possible. This monograph
has been written in the classical spirit of Fricke and Klein ([31]) and in that
of Lehner ([57]). Our main goal is to see how far the concrete computations
based on uniformization take us. It turns out that this simple approach
leads to a rich theory and opens a new way of treating the moduli problem.
Or rather puts new life in the classical methods that were used in the study
of moduli problems already in the 1920’s.
Some results, like the Uniformization of Riemann surfaces, have to be
presented here without proofs. They are, however, used almost exclusively
to interpret the results derived by other means. Proofs are not really based
on them. In all cases, where we do not present proofs, we furnish exact
references.
If one is willing to accept Uniformization and some related facts, then
this monograph is self-contained and can be read without much prior knowl-
edge about complex analysis.
In Chapter 1 we develop an engine that will power other chapters. There
we consider Mobius transformations and matrices. One of our aims in Chap-
ter 1 is to understand thoroughly how commutators of Mobius transforma-
tions behave and how groups generated by Mobius transformations can be
parametrized. All considerations here are elementary, but sometimes tech-
nically complicated.
In Chapter 2 we present some basic results of the theory of quasiconfor-
3
4 INTRODUCTION
ma1 mappings. Everything there is presented without proofs, which can be
found in the monograph of Lars V. Ahlfors [6] and in that of Olli Lehto and
Kalle Virtanen [59]. Quasiconformal mappings have played an important
role in the theory of Teichmiiller spaces. They provided the tools with which
it was possible to develop the first rigorous treatment of the moduli prob-
lem. Today most of the results concerning Teichmiiller spaces and moduli
spaces can be shown even without quasiconformal mappings. Quasiconfor-
ma1 mappings are only absolutely necessary to show that the moduli space
of symmetric Riemann surfaces of a given topological type is connected (cf.
Theorem 4.4.1 on page 147).
In Chapter 3 we first review the Uniformization of Riemann surfaces
without proofs. Then we show how considerations of Chapter 1 can be ap-
plied to study the geometry of Riemann surfaces. Our main concern in this
Chapter is to study the geometry of hyperbolic metrics of Riemann surfaces
of negative Euler characteristics. We derive many results concerning simple
closed geodesics and sizes of collars around them. We pay special attention
to the geometry of symmetric Riemann surfaces, i.e., to non-classical Klein
surfaces.
Everything here can be shown in detail using the results of Chapter 1. It
is actually surprising how much information can be obtained from detailed
analysis of the commutator of Mobius transformations. Considerations of
Chapter 3 form a quite comprehensive treatment of certain aspects of the
geometry of hyperbolic surfaces. So it may be of some interest for its own
sake already. Main target is, however, to get information about the moduli
problem using considerations of Chapter 1 alone. The beginning of Chapter
3 provides an environment in which considerations of Chapter 1 can be
interpreted so that we get useful results for later applications.
In Chapter 4 we introduce Teichmuller spaces and define its topology
using quasiconformal mappings. Here we have to resort to the review pre-
sented without proofs in Chapter 2. We will, however, later derive an alter-
native way of parametrizing the Teichmuller space using the geodesic length
functions. That is done in detail here (see page 161).
Quasiconformal mappings provide a simple way to describe the complex
structure of the Teichmuller space of classical Riemann surfaces (cf. page
148). We will take benefit of that description and indicate how our consid-
erations lead to a real analytic theory of Teichmuller spaces. This also leads
to a presentation of Teichmiiller spaces as a component of an affine real al-
gebraic variety (Section 4.12). This affine structure is derived here in detail.
This is an important part in the theory of Teichmiiller spaces, albeit not
central, because it opens new ways of compactifying the Teichmiiller space
by using methods of real algebraic geometry (cf. 1641, [16], [73]). We will not
consider these interesting approaches to the compactification problem here.
INTRODUCTION 5
Figure 0.1: The Mobius strip and the Klein bottle are two genus 1 real alge-
braic curves that are not homeomorphic to each other. In this monograph a
new moduli space is constructed for the these non-classical Klein surfaces.
The presentation of the affine structure of Teichmuller spaces is, however,
partly motivated by these new applications of real algebraic geometry.
Points of the moduli space of compact genus g Riemann surfaces are
isomorphism classes of mutually homeomorphic genus g Riemann surfaces.
Such Riemann surfaces are smooth projective complex algebraic curves. So
the moduli space of genus g Riemann surfaces is the same thing as the
moduli space of smooth genus g complex algebraic curves.
In Chapter 5 we consider this moduli space and define a natural topol-
ogy for it. The definition of the topology is based on the Fenchel-Nielsen
coordinates. In that topology the moduli space is connected but not com-
pact. Using the considerations of Chapter 3 we then consider degenerating
sequences of Riemann surfaces. It turns out that by adding points corre-
sponding to so called stable Riemann surfaces it is possible to compactify
the moduli space of compact and smooth genus g Riemann surfaces. This
is quite classical today and has first been shown by David Mumford and
others using the methods of complex algebraic geometry.
Smooth projective real algebraic curves have more structure than com-
plex curves. They can be viewed as compact Riemann surfaces with sym-
metry. Equally well they can be viewed as compact non-classical Klein
surfaces, i.e., surfaces that are obtained as the quotient of a smooth Rie-
mann surface by the action of the symmetry. This fact was realized already
by Felix Klein (cf. [46]). Therefore the inoduli spaces of non-classical com-
pact Riemann surfaces are simply moduli spaces of real algebraic curves.
A compact genus g surfaces has L(3g +4)/2J topologically different orien-
tation reversing symmetries. It follows, especially, that real algebraic curves
of the same genus need not be homeomorphic to each other. This implies
that, in any reasonable topology, the moduli space of smooth genus g real