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Advanced Lectures in Mathematics (ALM) Advanced Lectures in Mathematics (ALM) ALM 1: Superstring Theory ALM 2: Asymptotic Theory in Probability and Statistics with Applications ALM 3: Computational Conformal Geometry ALM 4: Variational Principles for Discrete Surfaces ALM 6: Geometry, Analysis and Topology of Discrete Groups ALM 7: Handbook of Geometric Analysis, No. 1 ALM 8: Recent Developments in Algebra and Related Areas ALM 9: Automorphic Forms and the Langlands Program ALM 10: Trends in Partial Differential Equations ALM 11: Recent Advances in Geometric Analysis ALM 12: Cohomology of Groups and Algebraic K-theory ALM 13: Handbook of Geometric Analysis, No. 2 ALM 14: Handbook of Geometric Analysis, No. 3 ALM 15: An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices ALM 16: Transformation Groups and Moduli Spaces of Curves ALM 17: Geometry and Analysis, No. 1 ALM 18: Geometry and Analysis, No. 2 ALM 19: Arithmetic Geometry and Automorphic Forms ALM 20: Surveys in Geometric Analysis and Relativity ALM 21: Advances in Geometric Analysis ALM 22: Differential Geometry: Under the Influence of S.-S. Chern ALM 23: Recent Developments in Geometry and Analysis Advanced Lectures in Mathematics Volume XXIII Recent Developments in Geometry and Analysis edited by Yuxin Dong · Jixiang Fu · Guozhen Lu Weimin Sheng · Xiaohua Zhu International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XXIII Recent Developments in Geometry and Analysis Volume Editors: Yuxin Dong (Fudan University) Jixiang Fu (Fudan University) Guozhen Lu (Wayne State University) Weimin Sheng (Zhejiang University) Xiaohua Zhu (Peking University) Copyright © 2012 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with the article.) In such cases, requests for permission to use or reprint should be addressed directly to the author. ISBN: 978-1-57146-250-3 Printed in the United States of America. 16 15 14 13 12 1 2 3 4 5 6 7 8 9 ADVANCED LECTURES IN MATHEMATICS Executive Editors Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor Editorial Board Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China Lo Yang Zhouping Xin Institute of Mathematics The Chinese University of Hong Kong Chinese Academy of Sciences (CAS) Hong Kong, China Beijing, China Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Sun Yat-sen University Guangzhou, China Dedicated to Professors Zheng-guo Bai and Yibing Shen Preface This is a special volume dedicated to Professor Zhengguo Bai (aka Chenkuo Pa) on the occasion of his 95th birthday and to Professor Yibing Shen on the occasion of his 70th birthday. As friends and students of Professors Bai and Shen, it is our great pleasure to edit this volume to celebrate their mathematical lives and careers. In addition to their own important mathematical contributions in projective geometry, Riemannian and differential geometry and Finsler geometry, Professors Bai and Shen have made tremendous contributions in educating and training of geometers in China. Professors Bai and Shen have trained around 80 M.S. and Ph.D. students over the years, including over 60 students jointly supervised by them. We refer the reader to “A list of graduate students of Professors Bai and Shen”. We also refer the reader to “A brief introduction of Professor Zhengguo Bai” and “A brief introduction of Professor Yibing Shen” for more information about their mathematical lives. This volumes describes some recent development in differential geometry and geometric analysis. Papers in this volume address a wide array of research topics. We are confident that this volume will be useful for geometers and analysts and mathematicians working in related areas. In particular, it can be very beneficial to graduate students, postdoctors, and young mathematicians who are just beginning their research in these areas. We would like to take this opportunity to thank wholeheartedly contributors of papers in this volume. This volume cannot be in the present form without their hard work and mathematical contributions. We like to thank C. Sogge, Editor-in-Chief of American Journal of Mathematics, as well for his short tribute to Professor Zhengguo Bai who published four papers in the American Journal of Mathematics in the 1940’s. We also thank Editors of the series of Advanced Lectures in the Mathematics for their support in the process of editing of this volume, in particular to Lizhen Ji for his initial invitation and solicitation to have this volume in honor of Professors Bai and Shen, and his constant encouragement and valuable suggestions. Special thanks are also due to Ms. Liping Wang and Ms. Huaying Li from the Higher Education Press for their tireless support and hard work. Without the devotion of Ms. Wang and Ms. Li, we could not have had completed this well organized volume in due time. Lastly, it is our great honor to be in charge of editing this volume in cele- bration of the mathematical lives of Professors Bai and Shen. As former students and friends of Professors Bai and Shen, we would like to take this opportunity to express our own sincere appreciation to them for their constant teaching, support to our mathematical careers, their long-lasting friendship. We truly admire their mathematical contributions and the enormous impact of their training of differ- ii Preface ential geometers, who have become by now a major composition of researchers in these ares in China. Yuxin Dong, Fudan University, China Jixiang Fu, Fudan University, China Guozhen Lu, Wayne State University, USA Weimin Sheng, Zhejiang University, China Xiaohua Zhu, Peking University, China May, 2012 A Tribute by Christopher Sogge to Professor Zhengguo Bai Professor Zhengguo Bai grew up in difficult times from a humble background and was largely self-taught. Unlike most other Chinese mathematicians of his time he did not study abroad. He did not receive a Ph.D., but, on the basis of his research, was promoted to Professor at Zhejiang University and later became the Department Chair of Hangzhou University. The war with Japan greatly affected Zhejiang University and caused it to be moved from Hangzhou. Afterwards, Pro- fessor Bai was one of the top leaders responsible for the revival of Hangzhou as a top center of mathematics. He also was an excellent and dedicated teacher who trained dozens of important Chinese geometers. Professor Zhengguo Bai also published papers at the beginning of his career under the name of Chenkuo Pa. These included many in top mathematical jour- nals. Among these are several that were published in the American Journal of Mathematics in the 1940’s. The latter include an important series of papers in the beginning of that decade that solved a famous problem Fubini, which followed earlier work of Blaschke, in projective differential geometry and ruled surfaces. He also studied projective minimal surfaces of coincidence, including quadrics of Boutard, Gordeaux sequences of quadrics, which resulted in other papers pub- lished in the American Journal of Mathematics. The papers speak for themselves, but, at that time, it was rare for Chinese mathematicians to be publishing in international journals of that caliber. As the Editor-in-Chief of the American Journal of Mathematics and someone with close ties to Zhejiang University, it is an honor to write this short tribute to Professor Bai who has contributed so much both through his research articles which were published in our journal and many other top journals and through his scholarship and teaching. Christopher Sogge1 Baltimore, Maryland, USA May 27, 2011 1Christopher Sogge is J. Sylvester Professor at the John Hopkins University. He was an invited speaker of the ICM in 1994, a Presidential Young Investigator Award recipient in 1989 and received a Guggenheim Fellowship in 2005. He is currently an overseas distinguished professor of Zhejiang University awarded by the Minstry of Educations of China. He has been the Editor- in-Chief of the American Journal of Mathematics since 2005. A Brief Introduction of Professor Zhengguo Bai (aka Chenkuo Pa) Professor Bai was born on December 17, 1916. His father managed a small phar- macy in the town to support the family. His father died when Bai was 3. As a result, his mother had to move with him to a village. Bai attended the elementary school in the village when he was 8. The whole school had only one teacher and Bai was mainly taught Chinese and literature. When Bai was in the third grade, the school hired a mathematics teacher and Bai started to learn mathematics. Bai was highly praised by his mathematics teacher for his mathematical talent. In 1933, Bai graduated from a high school. He was prepared to quit further school training because of his family financial difficulty and found a job as an elemen- tary school teacher.
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