2 1. RICCI SOLITONS
Some 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
functional 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 g(t) of the Ricci flow
on
Mn
is a Ricci soliton (or self-similar solution) if there exist a positive
function a(t) and a 1-parameter family of diffeomorphisms (p(t) : M M
such that
(1.1) g(t) = a(tMt)*g(0).
Let dJltt denote the space of Riemannian metrics on a differentiable man-
ifold M, and let 2)iff denote the group of diffeomorphisms of M. Consider
the quotient map IT : OKet 9DTet/2Hff x R+, where E
+
acts by scalings.
One verifies that a Ricci soliton is a solution g (t) of the Ricci flow for which
n (g (£)) 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) - 2 Re (g(t)) = &(t)(p(t)*go + r(t)ip(t)* (Cxgo),
where go ?(0), C denotes the Lie derivative, X is the time-dependent
vector field such that X (p(i) (p)) = ^ (ip(t)(p)) for any p G M, and fr = 4*.
DEFINITION
1.2. For obvious reasons, we say that g(t) is expanding,
steady, or shrinking at a time £o if & (^o) is 0, 0, or 0, respectively.
Since Rc(g(t)) = ^(£)*Rc(go)j w e c a n drop the pullbacks in (1.2) and
get
(1.3) -2Rc(#
0
) = cr{t)go + £x(t)#o,
where X (t) = a(t)X (t). Although go is independent of time, both &(t) and
X (t) may depend on time. For example, static Euclidean space
(Mn,
g{t)) =
(Kn,pCan)
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 a (t) 0 with J(0) = 1, consider the diffeomorphisms
tp(t) :
Rn
-
Rn
defined by (p(t) (x) =
a{t)~l/2x
for x e
W1.
Then
^(^)*^can =
cr(0_1^can.
Since g(t) = ^can, we may rewrite this as
(1.4) 9(t)=gCa* = (r(tMt)*g{0).
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