Perelman's work is based. In view of the desired topological applications of
Ricci flow, in Chapter 9 we give a more detailed review of 3-manifold topol-
ogy than was presented in Volume One. We hope to discuss the topics of
nonsingular solutions (and their variants), where one can infer the existence
of a geometric decomposition, surgery techniques, and more advanced topics
in the understanding of singularities
The second part of this volume emphasizes analytic and geometric tech-
niques which are useful in the study of Ricci flow, again especially in regards
to singularity analysis. We hope that the second part of this volume will
not only be helpful for those wishing to understand analytic and geometric
aspects of Ricci flow but that it will also provide tools for understanding
certain technical aspects of Ricci flow. The appendices form an eclectic col-
lection of topics which either further develop or support directions in this
We have endeavored to make each of the chapters as self-contained as
possible. In this way it is hoped that this volume may be used not only
as a text for self-study, but also as a reference for those who would like to
learn any of the particular topics in Ricci flow. To aid the reader, we have
included a detailed guide to the chapters and appendices of this volume and
in the first appendix we have also collected the most relevant results from
Volume One for handy reference.
For the reader who would like to learn more about details of Perelman's
work on Hamilton's program, we suggest the following excellent sources:
Kleiner and Lott [231], Sesum, Tian, and Wang [326], Morgan and Tian
[273], Chen and Zhu [81], Cao and Zhu [56], and Topping [356]. For further
expository accounts, please see Anderson [5], Ding [126], Milnor [267], and
Morgan [272]. Part of the discussion of Perelman's work in this volume was
derived from notes of four of the authors [102].
Finally a word about notation; if an unnumbered formula appears on
p. ^ ^ of Volume One, we refer to it as (Vl-p. ^?4b); if the equation is num-
bered )••, then we refer to it as (Vl-0.J|fc).
Highlights of Part I
In Part I of this volume we continue to lay the foundations of Ricci
flow and give more geometric applications. We also discuss some aspects of
Perelman's work on the Ricci flow.4 Some highlights of Part I of this volume
are the following:
(1) Proof of the existence of the Bryant steady soliton and rotationally
symmetric expanding gradient Ricci solitons. Examples of homoge-
neous Ricci solitons. Triviality of breather solutions (no nontrivial
steady or expanding breathers result). The Buscher duality trans-
formation of gradient Ricci solitons of warped product type. An
Some of these topics will appear in Part II of this volume.
Further treatment of Perelman's work will appear in Part II and elsewhere.
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