Table Of ContentBrownian Motion and Index
Formulas for the de Rham Complex
Kazuaki Taira
Institute of Mathematics
University of Tsukuba
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In this book it is studied how differential operators on a smooth man-
ifold reveal deep relationships between the topology of the manifold and
analysis on the manifold. The purpose is an introduction to central topics in
analysis on manifolds through the study of Laplacian-type operators on man-
ifolds. The main subjects covered are the celebrated Hodge-Kodaira theory
for Laplacians on forms in detail and also the Atiyah-Singer index theory in
less detail. This is a broad and active area of research, and has been treated
in advanced research monographs. In this book in contrast we give an ana-
lytic proof of an index formula for the relative de Rham cohomology groups
which may be considered as a generalization of the Hodge-Kodaira theory for
the absolute de Rham cohomology groups. Unlike many other books on in-
dex formulas in differential geometry, this book focuses on close relationships
between Markov processes and the de Rham complex, with emphasis on the
study of elliptic boundary value problems. Our approach is distinguished by
the extensive use of the techniques characteristic of recent developments in
the theory of linear partial differential equations. The main technique used
is the calculus of pseudo-differential operators on the manifold. The crucial
point is how to find an operator for which an index formula holds true in the
framework of spaces of currents. In deriving our index formula, the theory
of harmonic forms satisfying an interior boundary condition plays a funda-
mental role. Several recent developments in the theory of partial differential
equations have made possible further progress in the study of boundary value
problems and hence the study of index formulas for the de Rham complex.
The presentation of these new results is the main purpose of this book. Our
approach has a great advantage of intuitive interpretation of the index for-
mula in terms of Brownian motion from the point of view of probability
theory. Our results may be stated as follows: Brownian motion describes
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8 Preface
the topology of a compact Riemannian manifold through its Euler-Poincar´e
characteristic.
This book grew out of lecture notes for graduate courses given by the
author at the University of Tsukuba in 1994–1995 and at the Hiroshima
University in 1996–1998. It is addressed to graduate students and mathe-
maticianswithaninterestindifferentialgeometry,functionalanalysis,partial
differential equations and probability.
The contents of the book are divided into four principal parts.
The first part (Chapters 1–4) provides the elements of differential geom-
etry, functional analysis, Markov processes and partial differential equations
which are used throughout the book. In particular we summarize basic defi-
nitions and results about Sobolev spaces which enter naturally in connection
with interior boundary value problems. These function spaces play an im-
portant role in formulating the exterior derivative d in terms of spaces of
currents in Chapters 6 and 7. Furthermore the theory of pseudo-differential
operators – a modern theory of potentials – forms a most convenient tool in
the study of elliptic boundary value problems in Chapter 7. The material in
these preparatory chapters is given for completeness, to minimize the neces-
sity of consulting too many outside references. This makes the book fairly
self-contained.
Our subject proper starts with the second part (Chapter 5) where we
formulate various index formulas for the de Rham complex, and further give
anintuitive interpretation to theindexformulas interms ofBrownian motion
from the point of view of probability theory. In particular, the underlying
probabilistic mechanism of index formulas is revealed here. This plays an
important role in the interpretation and study of our index formula for the
relative de Rham cohomology groups.
The third part (Chapters 6–8) is devoted to interior elliptic boundary
value problems in the framework of spaces of currents by using the calculus
of pseudo-differential operators. Our approach is, in spirit, not far removed
from the classical potential approach.
The fourth and final part (Chapters 9–10) is devoted to the proof of our
index formula.
To make the material in Chapters 6–10 accessible to a broad spectrum of
readers, we have added Introduction. In this introductory chapter, we have
included two elementary (but important) examples of our index formula.
Furthermore we have attempted to state our problem and results in such
a fashion that a broad spectrum of readers could understand, and also to
Preface 9
describe how the problem can be solved, using the mathematics presented in
Chapters 1–4.
The book will be of great appeal to both advanced students and re-
searchers. For the former, it may serve as an effective introduction to three
interrelated fields of mathematics: differential geometry, elliptic boundary
value problems and Markov processes. For the latter, it provides a method
for the analysis of index formulas from the point of view of probability the-
ory, a powerful method clearly capable of extensive further development. We
hope that this book will lead to a better insight into the study of elliptic dif-
ferentialoperatorsofsecondorder,andfurtherthatthereaderwillappreciate
intimate connections between differential geometry and Markov processes.
This work was begun at the Institute of Mathematics of the University
of Potsdam, Germany, in February 1996 while I was participating in the
Spring School “Pseudo-Differential Calculus and Index Theory on Singular
and Non-Compact Manifolds”. In preparing this book, I am indebted to
many colleagues and friends. In particular, I would like to thank Profes-
sors Daisuke Fujiwara and Koˆichi Uchiyama for their constant interest in
my work. I am grateful to Professor Izumi Kubo for fruitful conversations
on the interpretation of our index formulas in terms of Brownian motion.
He also helped me to prepare the manuscript in a camera-ready form using
AMS-TEX. I would like to extend my warmest thanks to Professor Elmar
Schrohe who suggested the publication in its present form. I also thank the
referee who made many comments which were crucial in improving the expo-
sition, both mathematically and stylistically. My special thanks go to Mrs.
Gesine Reiher of WILEY-VCH Verlag Berlin for her unfailing helpfulness
and cooperation during the production of the book.
This research was partially supported by Grant-in-Aid for General Sci-
entific Research (No. 10440050), Ministry of Education, Science and Culture
of Japan.
Hiroshima and Tsukuba, August 1998
Kazuaki Taira
ABSTRACT
Thepurposeofthismonographistogiveananalytic proofofanindexfor-
mula for the relative de Rham cohomology groups which may be considered
as a generalization of the celebrated Hodge-Kodaira theory for the absolute
de Rham cohomology groups. More precisely, let X be a compact oriented
smooth Riemannian manifold without boundary, and Y a submanifold of X.
Our purpose is to find an operator D such that indD = χ(X)−χ(Y) where
χ(X) and χ(Y) are the Euler-Poincar´e characteristics of X and Y, respec-
tively. The crucial point is how to introduce spaces of currents on X and Y
in which the index formula for D holds true. In deriving our index formula,
the theory of harmonic forms satisfying an interior boundary condition plays
a fundamental role. Our approach has a great advantage of intuitive inter-
pretation of the index formula in terms of Brownian motion from the point
of view of probability theory. Our result may be stated as follows: Brownian
motion describes the topology of a compact Riemannian manifold through
its Euler-Poincar´e characteristic.
1991 Mathematics Subject Classification: Primary 58A12, 58G10; Sec-
ondary 35J25, 60J60.
Key words and phrases: relative de Rham cohomology group, Euler-
Poincar´e characteristic, index formula, interior elliptic boundary value prob-
lem, Brownian motion.
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ACKNOWLEDGMENTS
This work was begun at the Institute of Mathematics of the University of
Potsdam, Germany, in February 1996 while I was participating in the Spring
School“Pseudo-DifferentialCalculusandIndexTheoryonSingularandNon-
Compact Manifolds”. A part of the work was done at the University of Bari,
Italy, in May 1997 in the course of the Japan Society for the Promotion of
Science (JSPS) and the Italian “Consiglio Nazionale delle Ricerche” (CNR)
exchange program while I was on leave from Hiroshima University. I would
like to take this opportunity to express my gratitude to all these institutions
for their hospitality and support.
This research was partially supported by Grant-in-Aid for General Scien-
tific Research (No. 10440050), Ministry of Education, Science and Culture.
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Introduction 13
Chapter 1 Elements of Differential Geometry 33
1.1 Tangent Bundles 33
1.2 Vector Fields 35
1.3 Cotangent Bundles 39
1.4 Tensors 40
1.5 Tensors Fields 42
1.6 Exterior Product 43
1.7 Differential Forms 46
1.8 The de Rham Complex 48
1.9 The Codifferential, Hodge Star and Laplace-Beltrami
Operators 49
Chapter 2 Elements of Functional Analysis 55
2.1 Transpose Operators 55
2.2 The Riesz Representation Theorem 56
2.3 Closed Operators 58
2.4 Compact Operators 59
2.5 The Riesz–Schauder Theory 59
2.6 Fredholm Operators 61
2.7 Adjoint Operators 62
2.8 The Hilbert–Schmidt Theory 64
2.9 Theory of Semigroups 65
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12 Table of Contents
Chapter 3 Elements of Markov Processes 69
3.1 Conditional Probabilities 69
3.2 Brownian Motion 70
3.3 Markov Processes 71
3.4 Markov Transition Functions and Feller Semigroups 73
3.5 Theory of Feller Semigroups 78
Chapter 4 Elements of Partial Differential Equa-
tions 85
4.1 Sobolev Spaces 85
4.2 Fourier Integral Operators 90
4.3 Pseudo-Differential Operators 96
4.4 Pseudo-Differential Operators on a Manifold 101
4.5 EllipticPseudo-DifferentialOperatorsandtheirIndices 103
4.6 Potentials and Pseudo-Differential Operators 115
4.7 Spaces of Currents 118
Chapter 5 Index Formulas for the de Rham Com-
plex 121
5.1 The Boundaryless Case 121
5.2 The Bounded Case 127
Chapter 6 The Hodge–Kodaira Decomposition
Theorem 141
Chapter 7 The Exterior Derivative and the Co-
differential Operator 147
7.1 Elementary Formulas 147
∗
7.2 The Operators d and d 152
7.3 The Relative Hodge–Kodaira Decomposition Theorem 166
7.4 The Hodge–Kodaira Decomposition Theorem with
Boundary Condition 173
Chapter 8 The Operator D 179
Chapter 9 The Long Exact Sequence and the Op-
erator D 187
Chapter 10 Proof of Theorem 9.3 195
Table of Contents 13
Bibliography 207
Subject Index 211
