2 1. RICCI SOLITONSSome highlights of this chapter are a systematic description of the equa-
tions obtained from differentiating the gradient Ricci soliton equation, the
constructions of the Bryant steady Ricci soliton and the rotationally sym-
metric expanding Ricci soliton with positive curvature operator, examples of
homogeneous Ricci solitons, introduction to Perelman's energy and entropy
functionals via gradient Ricci solitons, and the Buscher duality transforma-
tion.1. General solitons and their canonical forms
We begin by recalling the following.DEFINITION 1.1 (General Ricci soliton). A solution 9(t) of the Ricci flow
on Mn is a Ricci soliton (or self-similar solution) if there exist a positive
function O"(t) and a I-parameter family of diffeomorphisms rp(t) : M --+ M
such that
(1.1) 9(t) = O"(t)rp(t)*9(0).Let 9Jtet denote the space of Riemannian metrics on a differentiable man-ifold M, and let :Diff denote the group of diffeomorphisms of M. Consider
the quotient map 7r : 9Jtet--+ 9Jtet/:Diff x IR+, where IR+ acts by scalings.
One verifies that a Ricci soliton is a solution 9 (t) of the Ricci flow for which
7r (9 ( t)) is independent of t, i.e., stationary.
We start by looking at what initial conditions give rise to Ricci solitons.
Differentiating ( 1.1) yields(1.2) -2Rc (9(t)) = &(t)rp(t)*90 + O"(t)rp(t)* (.Cx90),
where 90 = 9(0), .C denotes the Lie derivative, X is the time-dependent
vector field such that X (rp(t) (p)) = ft (rp(t)(p)) for any p EM, and & ~ ~~.DEFINITION 1.2. For obvious reasons, we say that 9(t) is expanding,
steady, or shrinking at a time to if & (to) is > 0, = 0, or < 0, respectively.Since Rc(9(t)) = rp(t)* Rc(90), we can drop the pullbacks in (1.2) and
get(1.3)where X (t) = O"(t)X (t). Although 9o is independent of time, both &(t) and
X (t) may depend on time. For example, static Euclidean space (Mn, 9(t)) =
(IRn, 9can) is a stationary solution to the Ricci flow and, as such, is a steady
Ricci soliton; however, this solution may also be considered as a Ricci
soliton which expands or shrinks modulo diffeomorphisms. In particular,
given any function O" (t) > 0 with O"(O) = 1, consider the diffeomorphisms
rp(t) : :!Rn -> :!Rn defined by rp(t) (x) = O"(t)-^112 x for x E :!Rn. Then
rp(t)*9can = O"(t)-^1 9can· Since 9(t) = 9can, we may rewrite this as
(1.4) 9(t) = 9can = O"(t)rp(t)*9(0).