ACKNOWLEDGMENTS xiii

(7) Applications of the reduced distance. Reduced volume of a static

metric and its monotonicity. Monotonicity formula for the re-

duced volume and application to weakened no local collapsing for

complete (possibly noncompact) solutions of the Ricci flow with

bounded curvature. Certain backward limits of ancient ^-solutions

are shrinking gradient Ricci solitons.

(8) A survey of basic 3-manifold topology and a brief description of the

role of Ricci flow as an approach to the geometrization conjecture.

(9) Concise summary of the contents of Volume One including some

main formulas and results. Formulas for the change in geometric

quantities given a variation of the metric, evolution of geometric

quantities under Ricci flow, maximum principles, curvature esti-

mates, classical singularity theory including applications of classical

monotonicity formulas, ancient 2-dimensional solutions, Hamilton's

partial classification of 3-dimensional finite time singularities.

(10) List of some results in the basic theory of Ricci flow and the back-

ground Riemannian geometry. Bishop-Gromov volume compari-

son theorem, Laplacian and Hessian comparison theorems, Calabi's

trick, geometry at infinity of gradient Ricci solitons, dimension re-

duction, properties of ancient solutions, existence of necks using

the combination of classical singularity theory in dimension 3 and

no local collapsing.

(11) Discussion of some results on the asymptotic behavior of complete

solutions of the Ricci flow on noncompact manifolds diffeomor-

phic to Euclidean space. A brief discussion of the mean curvature

flow (MCF) of hypersurfaces in Riemannian manifolds. Huisken's

monotonicity formula for MCF of hypersurfaces in Euclidean space,

including a generalization by Hamilton to MCF of hypersurfaces

in Riemannian manifolds. Short-time existence (Buckland) and

monotonicity formulas (Hamilton) for the cross curvature flow of

closed 3-manifolds with negative sectional curvature.

Acknowledgments

We would like to thank the following mathematicians for helpful dis-

cussions and/or encouragement: Sigurd Angenent, Robert Bryant, Esther

Cabezas-Rivas, Huai-Dong Cao, Xiaodong Cao, Jim Carlson, Albert Chau,

Bing-Long Chen, Xiuxiong Chen, Li-Tien Cheng, Shiu-Yuen Cheng, Yuxin

Dong, Klaus Ecker, David Ellwood, Bob Gulliver, Hongxin Guo, Emmanuel

Hebey, Gerhard Huisken, Bruce Kleiner, Brett Kotschwar, Junfang Li, Peter

Li, John Lott, Robert McCann, John Morgan, Andre Neves, Hugo Rossi,

Rick Schoen, Natasa Sesum, Weimin Sheng, Luen-Fai Tarn, Gang Tian, Pe-

ter Topping, Yuan-Long Xin, Nolan Wallach, Jiaping Wang, Guofang Wei,

Neshan Wickramasekera, Jon Wolfson, Deane Yang, Rugang Ye, Yu Yuan,

Qi Zhang, Yu Zheng, and Xi-Ping Zhu. We would like to thank Jiaping