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
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