GEOMETRIC ANALYSIS AROUND SCALAR CURVATURES

Transcription

1 GEOMETRIC ANALYSIS AROUND SCALAR CURVATURES

2 LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Chitat Chong and Kwok Pui Choi Institute for Mathematical Sciences National University of Singapore ISSN: Published Vol. 21 Vol. 22 Vol. 23 Vol. 24 Vol. 25 Vol. 26 Vol. 27 Vol. 28 Vol. 29 Vol. 30 Vol. 31 Environmental Hazards: The Fluid Dynamics and Geophysics of Extreme Events edited by H K Moffatt & E Shuckburgh Multiscale Modeling and Analysis for Materials Simulation edited by Weizhu Bao & Qiang Du Geometry, Topology and Dynamics of Character Varieties edited by W Goldman, C Series & S P Tan Complex Quantum Systems: Analysis of Large Coulomb Systems edited by Heinz Siedentop Infinity and Truth edited by Chitat Chong, Qi Feng, Theodore A Slaman & W Hugh Woodin Notes on Forcing Axioms by Stevo Todorcevic, edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang E-Recursion, Forcing and C*-Algebras edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Slicing the Turth: On the Computable and Reverse Mathematics of Combinatorial Principles by Denis R. Hirschfeldt, edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Forcing, Iterated Ultrapowers, and Turing Degrees edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Modular Representation Theory of Finite and p-adic Groups edited by Wee Teck Gan & Kai Meng Tan Geometric Analysis Around Scalar Curvatures edited by Fei Han, Xingwang Xu & Weiping Zhang *For the complete list of titles in this series, please go to

3 Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore Vol. 31 GEOMETRIC ANALYSIS AROUND SCALAR CURVATURES Editors Fei Han National University of Singapore, Singapore Xingwang Xu National University of Singapore, Singapore Weiping Zhang Nankai University, China World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI TOKYO

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Han, Fei (Mathematician), editor. Xu, Xingwang, 1957 editor. Zhang, Weiping, 1964 editor. Title: Geometric analysis around scalar curvatures / edited by Fei Han (NUS, Singapore), Xingwang Xu (NUS, Singapore), Weiping Zhang (Nankai University, China). Other titles: Lecture notes series (National University of Singapore. Institute for Mathematical Sciences) ; v. 31. Description: New Jersey : World Scientific, Series: Lecture notes series, Institute for Mathematical Sciences, National University of Singapore ; volume 31 Includes bibliographical references. Identifiers: LCCN ISBN (hardcover : alk. paper) Subjects: LCSH: Curvature. Geometric analysis. Riemannian manifolds. Algebraic topology. Classification: LCC QA645.G DDC 516.3/62--dc23 LC record available at British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2016 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

5 CONTENTS Foreword Preface Lectures on the Fourth-Order Q Curvature Equation Fengbo Hang and Paul C. Yang 1 An Introduction to the Finite and Infinite Dimensional Reduction Methods Manuel del Pino and Juncheng Wei 35 Einstein Constraint Equations on Riemannian Manifolds Qu ôc Anh Ngô 119 vii ix v

6 This page intentionally left blank

7 FOREWORD The Institute for Mathematical Sciences(IMS) at the National University of Singapore was established on 1 July Its mission is to foster mathematical research, both fundamental and multidisciplinary, particularly research that links mathematics to other efforts of human endeavor, and to nurture the growth of mathematical talent and expertise in research scientists, as well as to serve as a platform for research interaction between scientists in Singapore and the international scientific community. The Institute organizes thematic programs of longer duration and mathematical activities including workshops and public lectures. The program or workshop themes are selected from among areas at the forefront of current research in the mathematical sciences and their applications. EachvolumeoftheIMS Lecture Notes Series isacompendiumofpapers based on lectures or tutorials delivered at a program/workshop. It brings to the international research community original results or expository articles on a subject of current interest. These volumes also serve as a record of activities that took place at the IMS. We hope that through the regular publication of these Lecture Notes the Institute will achieve, in part, its objective of reaching out to the community of scholars in the promotion of research in the mathematical sciences. December 2015 Chitat Chong Kwok Pui Choi Series Editors vii

8 This page intentionally left blank

9 PREFACE During November 1 to December 31, 2014, the program Scalar Curvature in Manifold Topology and Conformal Geometry was held at the Institute for Mathematical Sciences (IMS) in the National University of Singapore (NUS). Curvatures are central concepts in differential geometry for understanding the shape of higher dimensional curved spaces. Scalar curvature, the simplest yet fundamental, has very deep and rich connections to analysis and topology on manifolds. The purpose of the program is to bring together researchers working on relevant areas to communicate ideas, unravel the connections and stimulate possible research collaborations. The program consisted of two international workshops and one winter school. The winter school was held during the week of December 2014 to train graduate students and young researchers. The speakers were either top experts or outstanding young researchers in the area. This volume collects three expanded lecture notes taught in this winter school. In the following, we shall give a brief overview. Chapter 1. Lectures on the Fourth-Order Q Curvature Equation by Fengbo Hang and Paul C. Yang (lecturer). In conformal geometry, a major tool is conformal covariant operators and their associated curvature invariants. The main purpose of this lecture is to introduce the fourthorder Paneitz operator and Q curvature, and their applications in understanding the conformal geometry and the topology of underlying manifold. Q-curvature is the quadratic analogue of the scalar curvature and Paneitz operator is the fourth-order type of conformal covariant operator similar to the conformal Laplace operator. These topics are recently developed and are still very active in conformal geometry. Chapter 2. An Introduction to the Finite and Infinite Dimensional Reduction Methods by Manuel del Pino and Juncheng Wei (lecturer). The authors introduce the finite dimensional and infinite dimensional Liapunov Schmidt reduction methods. These methods are illustrated by various model problems, especially in the infinite dimensional case, the Allen Cahn ix

10 x Preface equation and minimal surfaces equation. Current progress made by the authors demonstrated that these methods are very powerful in handling some nonlinear differential equations arising from geometry or physics. Chapter 3. Einstein Constraint Equations on Riemannian Manifolds by Qu ôc Anh Ngô. In the lecture notes, Ngô first briefly introduces the Einstein equation in general relativity theory. He then discusses the initial value problem and the derivation of Einstein constraint equations. In the last part, he illustrates how to apply the interesting slice method to study the Einstein constraint equations. We would like to thank all the participants of the program for their support and stimulating interactions during this successful two-month program. Our program was funded by IMS. We are indebted to Professor Chi Tat Chong, Director of IMS, for his guidance throughout the program and his leadership of IMS in providing us a relaxed but inspiring environment. Thanks also go to the IMS staff, especially to Emily Chan Ee Cheng and Eileen Tan whom, without their wonderful administrative support, the workshops would not have been as successful. November 2015 Fei Han National University of Singapore Singapore Xingwang Xu National University of Singapore Singapore Weiping Zhang Nankai University, China Volume Editors