ACKNOWLEDGMENTS xi will be used in Section 6 of Chapter 9. The Harnack estimates for surfaces are prototypes of those that apply in higher dimensions. We suggest that Chapter 6 be read in its entirety. In Chapter 7, read the statements of the main results. These give pre- cise insight into the smoothing properties of the Ricci flow and its long-time behavior. The Compactness Theorem stated in Section 3 is an essential in- gredient in important technique of analyzing singularity formation by taking limits of parabolic dilations. In Chapter 8, read Sections 1-3.1. These present the classification of maximal-time solutions to the flow, and introduce the method of parabolic dilation. In Chapter 9, read Section 1 for a heuristic description of singularity for- mation in 3-manifolds. Then read Sections 2 and 3 to gain an understanding of why positive curvature dominates near singularities in dimension three. Finally, review the results of Sections 4 and 6 to gain insight into the tech- nique of dimension reduction. Acknowledgments During the preparation of this volume, Bennett Chow received support from NSF grant DMS-9971891, while Dan Knopf was partially supported by DMS-0328233. B.C. would like to thank the following mathematicians from whom he has benefitted both mathematically and personally. He thanks Professor Shiing-Shen Chern whom he met in 1979, and who provided him with the inspiration to study differential geometry. From his graduate school days at Princeton and San Diego: Huai-Dong Cao, Craig Hodgson, Nikos Kapouleas, Igor Rivin, Fadi Twainy, and Cumrun Vafa. From his postdoc years at MIT, Rice, and NYU: Zhiyong Gao, Victor Guillemin, Richard Melrose, Louis Nirenberg, Dan Stroock, and Deane Yang. From the years at Min- nesota: his colleagues Scot Adams, Bob Gulliver, Conan Leung, Wei-Ming Ni, and Jiaping Wang, as well as his former students Sun-Chin (Michael) Chu, Alexei Krioukov, Lii-Perng Liou, Dan O'Loughlin, Dong-Ho Tsai, and Yu Yuan. From San Diego: his colleagues Li-Tien Cheng, Zheng-Xu (John) He, Lei Ni, and Nolan Wallach, as well as his former student David Glick- enstein. Special thanks to B.C.'s advisor, Professor Shing-Tung Yau, and teacher, Professor Richard Hamilton, for their invaluable teachings, support, and encouragement for more than 20 years, without which this book would not have been written. Very special thanks to Classic Dimension for encour- agement, support and inspiration. B.C. is indebted to his parents for the love, support, and encouragement they have given him all of these years, and to his father for teaching him mathematics beginning in his youth. B.C. ded- icates this book to his parents Yutze and Wanlin and to the memory of his sister Eleanor.
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