1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4 1. RICCI SOLITONS

A way to circumvent the undesirableness of the uniqueness assumption

in Proposition 1.3 is as follows. Since the geometry of any Ricci soliton

g(t) is the same as that of go, we will start with go and then construct

another Ricci soliton in canonical form and with-the same initial metric go.

Choose any time to and let E ~ 0-(to) and Xo ~ X(to), so that equation

(1.3) becomes at t =to

-2 Rc(go) = Ego + Cx 0 go.

We will now drop the subscripts on X and g. Using indices, the equation

above now reads

(1.8)

where Xi= gijXj are the components of Xb, the covariant tensor (1-form)

obtained from X by lowering indices using g. A triple (g, X, E) (or pair

(g, X) if we suppress the dependence on e) consisting of a metric and a

vector field that satisfies (1.8) for some constant Eis called a Ricci soliton

structure.^2 We say that X is the vector field the soliton is flowing along.

REMARK 1.5 (Solitons and normalized Ricci fl.ow). If (g, X) is a Ricci

soliton structure on a compact manifold M, then g evolves purely by dif-

feomorphisms under the normalized (constant volume) Ricci fl.ow.

DEFINITION 1.6 (Gradient Ricci soliton). A Ricci soliton structure (g, X)

is a gradient soliton structure if there exists a function f (called the po-

tential function) such that Xb = df. In this case, (1.8) becomes

(1.9)

The following, whose proof is elementary, shows that given a complete

gradient Ricci soliton structure, we can construct a gradient Ricci soliton

in canonical form (see Theorem 4.1 on p. 154 of [111] or Kleiner and Lott

[231]). In particular, the result below illustrates the sense in which a Ricci

soliton structure may be regarded as initial data for a Ricci solution, i.e., for

a self-similar solution to Ricci flow.

PROPOSITION l. 7 (Gradient soliton structures and canonical forms).

Suppose (go, \J f o, E) is a complete gradient Ricci soliton structure on Mn.

Then there exists a solution g ( t) of the Ricci flow with g ( 0) = g 0 , diffeomor-

phisms cp (t) with cp (0) = idM, and functions f (t) with f (0) = Jo defined

for all t with

(1.10) T (t) ~ 1 +Et> 0,

such that

(^2) Below, we will sometimes denote a Ricci soliton structure by (Mn,g,X), in order
to emphasize the underlying manifold, e.g. when M is a Lie group..