Spectral Geometry of the Laplacian: Spectral Analysis and Differential Geometry of the Laplacian

308 Pages·2017·2.303 MB·English

by  Hajime Urakawa

Spectral
Geometry
of the
Laplacian
Spectral Analysis and Differential
Geometry of the Laplacian
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Spectral
Geometry
of the
Laplacian
Spectral Analysis and Differential
Geometry of the Laplacian
Hajime Urakawa
Tohoku University, Japan
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
10018hc_9789813109087_tp.indd 2 21/10/16 9:22 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Urakawa, Hajime, 1946–
Title: Spectral geometry of the Laplacian : spectral analysis and differential geometry of the
Laplacian / by Hajime Urakawa (Tohoku University, Japan).
Description: New Jersey : World Scientific, 2017. | Includes bibliographical references.
Identifiers: LCCN 2016047370 | ISBN 9789813109087 (hardcover : alk. paper)
Subjects: LCSH: Symmetric matrices. | Laplacian. | Eigenvalues. |
Spectral geometry. | Riemannian manifolds.
Classification: LCC QA188 .U73 2017 | DDC 516.3/62--dc23
LC record available at https://lccn.loc.gov/2016047370
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
Printed in Singapore
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Preface
Preface to the Japanese version
This book aims to give a general theory of the spectral geometry of the
LaplacianonacompactRiemannianmanifold. Namely,wegivetheprecise
theory of the behavior of the eigenvalues and the eigenfunctions of the
Laplacian and the corresponding heat kernel acting on the space of C∞
functions on a compact Riemannian manifold.
ThehistoryoftheeigenvalueproblemoftheLaplacianwithfixedbound-
ary value is old, and is the basis of the mathematical sciences. Many im-
portant researches have been done since the works by Hermann Weyl in
1911. Recently, many important researches have been done related to the
theory of collapsing of Riemannian manifolds. We devote ourselves to give
the precise and details of the spectral theory of the Laplacian.
The contents of this book are as follows. Chapter 1 is the preparation
of the basic materials in Riemannian geometry which will be necessary in
the subsequent chapters. Chapter 2 states the basic and general properties
of the eigenvalues of the Laplacian which were studied by K. Uhlenbeck in
1976, and joint work with Shigetoshi Bando in 1983 which were recently
developed into the spectral theory of the Cauchy-Riemannian Laplacian
on Cauchy-Riemannian geometry by mathematicians from France, Italy,
USA and Japan including A. Aribi, A. El Soufi, S. Ilias, S. Dragomir and
E.M. Harrel.
In Chap. 3, we treat J. Cheeger and S.T. Yau’s works in 1975 on the
lower bounds of the least positive eigenvalues of the Laplacian. In partic-
ular, we give S.T. Yau’s estimation in terms of geometric quantities of the
curvature of Riemannian manifolds.
v
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vi Spectral Geometry of the Laplacian
In Chap. 4, we treat with S.Y. Cheng’s works on the upper estimations
of the kth eigenvalues of the Laplacian and R. Courant counting theorem
on the nodal sets and nodal domains. Moreover, we treat the theorem
due to A. Lichnerowicz and M. Obata to give the best lower bound of the
first eigenvalue of the Laplacian which has been applied in various field in
differential geometry.
In Chap. 5, we treat Q.M. Cheng’s recent works on Payne-P´olya-
Weinberger type inequalities of the eigenvalues of the Dirichlet eigenvalue
of the Laplacian on an arbitrary Riemannian manifold.
InChap.6,wetreatthebigworksduetoColindeVerdi`ere. Heshowed
theepoch-makingworksuchthatthespectrumoftheLaplaciandetermines
the totality of lengths of closed geodesics. In Chap. 6, we introduce the
detailsofhisbrillianttheory. HistheoryplaysanimportantrollinChap.7.
Professor Atsushi Katsuda in Kyushu University gave to me a detailed
5-page explanation of the proof of Colin de Verdi`ere’s theorem. In this
book, we have omitted J. Chazarain’s theory by using the wave equation
to show Colin de Verdi`ere’s theorem.
Chapter 7 is one of the highlights in this book which introduces the
works due to V. Guillemin and D. Kazhdan in ’80s. Their works are essen-
tially important on the spectral rigidity theorems of compact Riemannian
manifolds with negative curvature. We treat symplectic geometry, Anosov
flow and the geodesic flow.
Thisbookaredueto“SurveysinGeometry1980–81”,“Geometryofthe
Laplace Operator” by Takeshi Kotake, Yoshiaki Maeda, Shin Ozawa and
Hajime Urakawa except Chaps. 1 and 5. The author expresses his sincere
gratitude to Professor Emeritus Takushiro Ochiai at Tokyo University for
giving him a very kind opportunity to conduct his survey in the workshop
at Keio University, 1980, and also allowed him to write the Japanese ver-
sion of this book based on the survey note. Chapter 5 is due mainly to
Q.M. Cheng’s work. The author expresses his sincere gratitudes to Pro-
fessor Q.M. Cheng at Fukuoka University for quoting his works. Finally,
the author expresses his sincere gratitude to Professor Atsushi Katsuda at
Kyushu University for reading the manuscript of the Japanese version, for
correcting several mistakes and filling in the gaps in Chap. 6.
Hajime Urakawa at Sendai, Spring 2015
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Preface vii
Preface to the English version
We express our gratitude to Professor Emeritus Yusuke Sakane for his as-
sistanceinthefigures,andtoMs.ChionhEngHuayatWorldScientificCo.
Pte. Ltd. for the arrangements in the publication of the English version.
Hajime Urakawa at Sendai, Spring 2017
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