Contents of Part I of Volume Two
"There must be some way out of here, ..."
- Prom "All Along the Watchtower" by Bob Dylan
We describe the main topics considered in each of the chapters of Part
I of this volume.
Chapter 1. We consider the self-similar solutions to the Ricci flow,
called Ricci solitons. These special solutions evolve purely by homotheties
and diffeomorphisms. When the diffeomorphisms are generated by gradient
vector fields, the Ricci solitons are called gradient. We begin by presenting
a canonical form for gradient Ricci solitons and by systematically differenti-
ating the gradient Ricci soliton equations to obtain higher-order equations.
These equations play an important role in the qualitative study of gradient
Warped products provide elegant and important examples of solitons,
such as the cigar metric, which is an explicit steady Ricci soliton defined on
and which is conformal to the Euclidean metric, positively curved and
asymptotic to a cylinder. We also discuss the construction of the higher-
dimensional generalization of the cigar, the rotationally symmetric Bryant
soliton defined on
Interestingly, the qualitative behavior of the Bryant
soliton is quite different from the cigar. We also construct rotationally sym-
metric expanding gradient Ricci solitons.
An interesting class of Ricci solitons is the homogenous Ricci solitons.
It is notable that in dimensions as low as 3, there exist expanding homo-
geneous Ricci solitons which are not gradient. We especially discuss the
3-dimensional case of expanding Ricci solitons.
The scarcity of Ricci solitons on closed manifolds is exhibited by the
fact that the only steady or expanding Ricci solitons on such manifolds are
Einstein metrics. Moreover, in dimensions 2 and 3, the only shrinking Ricci
solitons on closed manifolds are Einstein. In dimension 2, this follows from
the Kazdan-Warner identity and in dimension 3 this relies on a pinching
estimate for the curvature due independently to Hamilton and Ivey.
The consideration of gradient Ricci solitons, in particular those geomet-
ric quantities which either vanish or are constant on gradient Ricci solitons,
has played an important role in the discovery of monotonicity formulas. We
briefly introduce Perelman's energy and entropy functionals from this point