x PREFACE other hand, we hope that experts in the field will appreciate the organization of the results, covering the span of more than a century, into a unified whole. Each chapter ends with bibliographical notes. Attributions are kept to a mini- mum in the body of the text, and the history and context of the works are expanded in the bibliographical notes. All works quoted herein are already published. Acknowledgments. We would like to express our gratitude to Shing-Tung Yau. Since the late 1970's, Yau has been promoting the problem of isometric embedding of Riemannian manifolds in Euclidean spaces. He taught a graduate course on this subject at the University of California at Berkeley in 1977 and at the National Tsing Hua University of Taiwan in 1989. His lecture notes were of interest among those studying the problem. It is fair to say that the research in this subject would not have reached its present level without his advocacy. It is with pleasure we record here our thanks to our thesis advisors, Chaohao Gu (for J.-X. H.) and Fanghua Lin (for Q. H.). Our mathematical careers began with their patient guidance many years ago. Their persistent encouragement played an essential role in our research. In preparing this book, we have benefitted greatly from discussions with many of our colleagues and friends. We especially thank Pengfei Guan, Yanyan Li and Chang-Shou Lin for many helpful discussions. A special thanks goes to Brian Smyth for reading the entire manuscript and for many suggested improvements. Part 2 was presented by Q. H. in a series of seminars when he was a visitor at the Max-Planck Institute for Mathematics in Leipzig in 2002-2003. He would like to thank MPI for its hospitality. In particular, he would like to thank Stefan Muller for arranging his visit and providing him with a good opportunity to write this book. Thanks also should be expressed to the American Mathematical Society, espe- cially Edward Dunne and Deborah Smith, for their invaluable assistance in bringing this book to press. The research related to this book was partially supported by a Sloan Foundation Fellowship and a grant from the National Science Foundation for Q. H. and a grant from the National Science Foundation of China for J.-X. H. Qing Han, Jia-Xing Hong
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