Teichmüller Theory in Riemannian Geometry

223 Pages·1992·15.067 MB·English

by  Anthony J. Tromba (auth.)

Lectures in Mathematics
ETHZiirich
Department of Mathematics
Research Institute of Mathematics
Managing Editor:
Oscar E. Lanford
Anthony J. Tromba
Teichmiiller Theory in
Riemannian Geometry
based on lecture notes by
Jochen Denzler
2nd revised printi ng
Springer Basel AG
Author's address:
Anthony J. Tromba
Mathematisches Institut
Ludwig-Maximilians-Universitiit Miinchen
Theresienstr. 39
D-8000 Miinchen 2
Anthony J. Tromba
Department of Mathematics
University of California
Santa Cruz, CA 95064
USA
Deutsche Bibliothek Cataloging-in-Publication Data
Iromba, Anthony J.:
Teichmiiller theory in Riemannian geometry: based on lecture notes by Jochen Denzler / Anthony
J. Tromba. - Springer Base! AG, 1992
(Lectures in mathematics)
ISBN 978-3-7643-2735-4 ISBN 978-3-0348-8613-0 (eBook)
DOI 10.1007/978-3-0348-8613-0
This work is subject to copyright. AII rights are reserved, whether the whole orpart ofthe material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of
the German Copyright Law where copies are made for other than private use a fee is payable to
«Verwertungsgesellschaft Wort», Munich.
© 1992 Springer Base! AG
Origina1ly published by Birkhiiuser Verlag in 1992
ISBN 978-3-7643-2735-4
98765432
Preface
These lecture notes are based on the joint work of the author and Arthur Fischer on
Teichmiiller theory undertaken in the years 1980-1986. Since then many of our colleagues
have encouraged us to publish our approach to the subject in a concise format, easily
accessible to a broad mathematical audience.
However, it was the invitation by the faculty of the ETH Ziirich to deliver the ETH
N achdiplom-Vorlesungen on this material which provided the opportunity for the author
to develop our research papers into a format suitable for mathematicians with a modest
background in differential geometry. We also hoped it would provide the basis for a
graduate course stressing the application of fundamental ideas in geometry. For this
opportunity the author wishes to thank Eduard Zehnder and Jiirgen Moser, acting director
and director of the Forschungsinstitut fiir Mathematik at the ETH, Gisbert Wiistholz,
responsible for the Nachdiplom Vorlesungen and the entire ETH faculty for their support
and warm hospitality.
This new approach to Teichmiiller theory presented here was undertaken for two reasons.
First, it was clear that the classical approach, using the theory of extremal quasi-conformal
mappings (in this approach we completely avoid the use of quasi-conformal maps) was not
easily applicable to the theory of minimal surfaces, a field of interest of the author over
many years. Second, many other active mathematicians, who at various times needed
some Teichmiiller theory, have found the classical approach inaccessible to them.
The spirit of this approach was partially inspired by a paper of Earle and Eells "On
a Fibre Bundle Description of Teichmiiller theory" published in 1969 in the Journal of
Differential Geometry, and is more in line with the traditional development of ideas in
geometry and partial differential equations. Moreover we intended to have the material
in this book, both the analytic as well as the geometric, reasonably self-contained.
Whereas various authors on classical Teichmiiller theory omit fundamental analytical
results like the existence and uniqueness of extremal quasi-conformal maps, we on the
other hand include (although in an appendix) the existence and uniqueness of harmonic
diffeomorphisms which form part of the analytical basis of this theory. We hope therefore
that these notes will indeed find their intended broad audience.
There are many individuals who have contributed to the existing literature in classical
Teichmiiller theory. Unfortunately due to the limitations of the Nachdiplom-Vorlesungen
we could not mention several of their important and interesting results, nor were we able
2
to present a dictionary between our approach and the classical one. Some of these results
are available in the books of Gardiner [40J and Lehto [65J.
In these lectures on Teichmiiller theory we develop the essentials of the subject from basic
fundamentals with the main intention of making Teichmiiller theory easy to learn. Our
readers must remain the final judges as to whether we succeeded in this goal. We should
also mention that there is some additional material included here that was not presented
in the original lectures.
Several people have helped us substantially in our efforts. First the Sonderforschungs
bereich 256 for Partial Differential Equations in Bonn under the direction of Stefan Hilde
brandt and the Max-Planck-Institute under the direction of Friedrich Hirzebruch very
generously supported the research which resulted in these notes. My thanks go to my
friend, colleague, and co-author Arthur Fischer who taught me much of the advanced
geometry I know.
Michael Buchner, Hans Duistermaat, Stefan Hildebrandt, Alan Huckleberry, Jerry Mars
den, Dick Palais, Andrey Todorov, and Friedrich Tomi provided encouragement as well
as mathematical inspirations. The proof of Poincare's theorem, section 1.5, and of the
collar lemma in the appendix are due to Tomi and the proof of the Mumford Compact
ness Theorem is due to Tomi and the author. Andreas Miiller provided the details of the
argument that 'Do is contractible at the end of section 3.4. Kurt Strebel, Ralph Strebel,
and Heiner Zieschang provided us with important historical information.
Our appreciation goes to our students and to Adimurthi, Horst Knorrer, Alfred Kiinzle,
Serge Lang, Michael Struwe, Eugene Trubowitz, and Eduard Zehnder, who attended
some or all of the lectures, and whose interest and comments added to the quality of our
presentation. We owe a great thanks to Yair Minsky who found an error in our original
approach to the Nielsen problem.
We also wish to thank Stefan Winiger for his careful typing of the manuscript and Artur
Barczyk for professionally drawing the pictures. This book, however, could never have
achieved its current polished form without the tireless and enthusiastic efforts of Jochen
Denzler for which the author is deeply appreciative.
A.J. Tromba
ETH Ziirich
June 1991
Contents
o Mathematical Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1 The Manifolds of Teichmiiller Theory
1.1 The Manifolds A and AS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 The Riemannian Manifolds M and ,IVls ......................... ....... 18
1.3 The Diffeomorphism M' Ips ~ AS ................. ........ ....... ..... 19
1.4 Some Differential Operators and their Adjoints. . . . . . . . . . . . . . . . . . . . . . . .. 26
1.5 Proof of Poincare's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
1.6 The Manifold M~l and the Diffeomorphism with MS Ips.......... .... 33
2 The Construction of Teichmiiller Space
2.1 A Rapid Course in Geodesic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
2.2 The Free Action of Vo on M -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
2.3 The Proper Action of Vo on M-1 ................................... .. 41
2.4 The Construction of Teichmiiller Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
2.5 The Principal Bundles of Teichmiiller Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 The Weil-Petersson Metric on T(M)................................... 60
4 Contents
3 T(M) is a Cell
3.1 Dirichlet's Energy on Teichmiiller Space ................................ 63
3.2 The Properness of Dirichlet's Energy. .... .... .. . .... ... .. . .. .. . ... . .. .. 74
3.3 Teichmiiller Space is a Cell... . . .. ... .. ....... .... . .. ... ... . . ... . ... .... 78
3.4 Topological Implications; The Contractibility of 'Do . . . . . . . . . . . . . . . . . . . .. 81
4 The Complex Structure on Teichmiiller Space
4.1 Almost Complex Principal Fibre Bundles.. .... . ... .... ... .. .. . ... . .. .. 83
4.2 Abresch-Fischer Holomorphic Coordinates for A. . . . .. . . .. .. . . . . . . . ... .. 90
4.3 Abresch-Fischer Holomorphic Coordinates for T(M}................... 94
5 Properties of the Weil-Petersson Metric
5.1 The Weil-Petersson Metric is Kahler.. .. ... . . .. . .. . . .. . .. .. . . . ... . ... .. 96
5.2 The Natural Algebraic Connection on A ............................... 102
5.3 Further Properties of the Algebraic Connection and the
non-Integrability of the Horizontal Distribution on A ................... 106
5.4 The Curvature of Teichmiiller Space with Respect to its
Weil-Petersson Metric ................................................. 111
5.5 An Asymptotic Property of Weil-Petersson Geodesics. . . . . . . . . . . . . . . . .. 121
6 The Pluri-Subharmonicity of Dirichlet's Energy on T(M);
T(M) is a Stein-Manifold
6.1 Pluri-Subharmonic Funct.ions and Complex Manifolds .................. 123
6.2 Dirichlet's Energy is Strictly Pluri-Subharmonic ........................ 126
6.3 Wolf's Form of Dirichlet's Energy on T(M) is Strictly
Weil-Petersson Convex ................................................. 138
6.4 The Nielsen Realization Problem ....................................... 152
Contents 5
A Proof of Lichnerowicz' Formula ......................................... 155
B On Harmonic Maps ...................................................... 158
C The Mumford Compactness Theorem .................................. 184
D Proof of the Collar Lemma .............................................. 192
E The Levi-Form of Dirichlet's Energy ............................ ....... 196
F Riemann-Roch and the Dimension of Teichmiiller Space ............. 201
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205
Indexes
Index of Notation ........................................................... 214
A Chart of the Maps Used .................................................. 218
Index of Key Words ......................................................... 219
o
Mathematical Preliminaries
Let us collect some definitions and facts from differential geometry, which will be useful
for our presentation:
Definition 0.1 A Coo n-manifold M (without boundary) is a paracompact topological Haus
dorff space, together with a maximal collection of open subsets (V;)iE! covering M: U Vi =
iEI
M, and homeomorphisms 'Pi : Vi ---> lRn such that whenever Vi n Vj =1= 0, 'Pi 0 'Pi! is Coo.
(The collection {(Vi, 'Pi)} has been suppressed in the notation here, as will be done, when no
confusion can arise.)
M is said to be orientable, if the covering can be chosen so that detD('Pi 0 'Pi!)('Pj(x)) is
always positive. If M is orientable, then, subject to this property, it has two possible maximal
coverings. A choice of one is called an orientation.
A Coo Banach manifold M is a paracompact topological Hausdorff space, together with
a maximal collection of open subsets (Vi)iE!, UiE! Vi = M, and of homeomorphisms
'Pi : Vi ---> E, where E is a Banach space, such that 'Pi 0 'Pi! is Coo whenever defined
(i.e. whenever Vi n Vj =1= 0).
If E is a Hilbert space, M is called a Coo Hilbert manifold.
Similarly, manifolds can be defined, which are modelled after any topological vector space
E (say e.g. a Frechet space), and which are of any differentiability class by stipulating
that 'Pi 'Pi! should be of the corresponding class. We are especially interested in
0
Definition 0.2 A Riema'nn surface M is a Coo oriented manifold, together with a collection
of local homeomorphisms 'Pi : Vi ---> <C in the given orientation with U Vi = M such that
'Pio'Pt : Vij ---> lR2 areholomorphic maps, where defined, i,e. if0 =1= Vii := 'Pi(VinVj) C lR2.
7
Here, IR? is to be considered as <C. The collection c := {(U;, 'Pi) liE I} is called a complex
structure for M. If M has a fixed complex structure, we write (M, c) to denote M with its
given complex structure.
Definition 0.3 A mapping f : M -+ N where (M, c), (N, c') are Riemann surfaces is
called holomorphic, iff for all charts (If;,,,pj) E c', (Ui, 'Pi) E c, the mapping "pj 0 f 0 'Pil is
holomorphic (where defined). A similar definition holds for Coo maps between Coo manifolds.
Note that the definition does not depend on the choice of the charts.
M
f
QUi
Figure 0.1: cf. definition 0.3
Definition 0.4
f : M -+ N is a diffeomorphism, iff f E Coo and f-1 is defined and E Coo
f : M -+ N is a holomorphic equivalence, iff both f and f-1 are holomorphic.
A classical result in surface theory [431 yields
Theorem 0.1 If M, N are compact, oriented Coo 2-manifolds (without boundary) which
are homeomorphic, then they are diffeomorphic.
This shows that the classification of compact oriented 2-manifolds without boundary
up to diffeomorphisms is the same as their classification up to homeomorphism, so the