1547671870-The_Ricci_Flow__Chow

22 2. SPECIAL AND LIMIT SOLUTIONS

solitons. We will study Ricci solitons in detail in a chapter of the successor
to this volume. But because these solutions often arise as limits of dilations
about singularities, we shall encounter some examples in this chapter. Hence
we provide a brief introduction to them here.
Suppose that (Mn,g (t)) is a solution of the unnormalized Ricci flow
on a time interval (a, w) containing 0, and set go = g ( 0). One says g ( t)
is a self-similar solution of the Ricci flow if there exist scalars CJ (t) and
diffeomorphisms 'l/Jt of Mn such that

(2.2) g ( t) = CJ ( t) 'I/;; (go)

for all t E (a, w). Thus a self-similar solution is a solution of the evolution
equation (2.1) having the special form (2.2). A metric having this form
changes only by diffeomorphism and rescaling. To see the significance of
this fact, let us denote by S2Mn the bundle of symmetric (2, 0)-tensors
on Mn and by st Mn the sub-bundle of positive-definite tensors. Then
the space of Riemannian metrics on Mn may be written as 9'J1 (Mn) ~
C^00 (Mn, st Mn). Let '.D (Mn) denote the diffeomorphism group of Mn.
Because the Ricci flow is invariant under diffeomorphism, it may be regarded
as a dynamical system on the moduli space

of Riemannian metrics modulo diffeomorphisms. In this context, Ricci soli-

tons correspond to generalized fixed points.

On the other hand, suppose that (Mn, go) is a fixed Riemannian mani-

fold such that the identity

(2.3) -2 Re (go)= .Cxgo + 2>..go

holds for some constant >.. and some complete vector field X on Mn. In this

case, we say go is a Ricci soliton. If X vanishes identically, a Ricci soliton

is simply an Einstein metric. Consequently, any solution of (2.3) (which

is actually a coupled elliptic system for go and X) may be regarded as a

generalization of an Einstein metric.

REMARK 2.1. By rescaling, one may assume that>.. E { -1, 0, 1} in (2.3).

These three cases correspond to shrinking, steady, and expanding soli-

tons, respectively. And as we shall see below, these yield examples of ancient,

eternal, and immortal solutions, respectively.

REMARK 2.2. In case the vector field X appearing in (2.3) is the gradient

field of a potential function -f, one has

V'V' f =Re +>..g

and says go is a gradient Ricci soliton.

A vector field X that makes a Riemannian metric into a Ricci soliton

may not be unique. The next example illustrates this.