1547671870-The_Ricci_Flow__Chow

4. UNIQUENESS OF RICCI SOLITONS

and its average scalar curvature by

--'- fMn Rdμ

P--c- V.

If g is a Ricci soliton, then (2.3) implies that

• 2 Re= Cxg + 2>.g,
  whence we get the divergence identity

n>.+R= 8X

by tracing. Integrating this identity shows that

(n>.+p)V=O,

117

hence that p = -n>.. It follows that the average scalar curvature of a soliton

that is respectively expanding, steady, or shrinking must be respectively

negative, zero, or positive. This is not surprising, because applying Lemma

3.9 to the corresponding self-similar solution implies that

d

dt log V ( t) = - p ( t).

PROPOSITION 5.20. Any expanding or steady self-similar solution of the

Ricci flow on a compact n-dimensional manifold is Einstein.

PROOF. We consider the corresponding solution (Mn,g(t)) of the nor-

malized Ricci fl.ow. (See Section 9 of Chapter 6.) Because g (t) flows by

diffeomorphisms, there exists a one-parameter family of 1-forms X (t) such

that

1 8 p

-2 otg = Rc-~g = Lx~9·

Thus by Corollary 6.64, we have

. 8 2 2p

• (\7 R, X) = -£xR = otR = ~R + 2 jRcl - -;;:R,

which implies that the identity

(5.15) ~R + (\7 R, X) +^2 jRcj^2 -

2

p R = 0

n

holds identically in time.

Now assume g (t) is an expanding or steady self-similar solution. Then

p (t) :::; 0 as we observed above. Replacing Re by Re -~g, one finds that

equation (5.15) is equivalent to

(5.16) ~ (R - p) + (\7 (R - p), X) + 2 IRc -~gl

2

+ ~ (R - p) = 0.

By (5.16), at any point x E Mn such that R (x, t) = Rmin (t), we have

2 !Rc-~ 91

2

+ ~ (R-p):::; o.