1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS

PROOF. By (30.120), we have

1 2

2Rc(X) + V'xX + 2\7 IXI - X = 0.

Thus

~x 1x1

2

= (\7 xx, X) = -2 Re (X, X) - ~x 1x1

2

+ 1x1

2

,

so that X IXl

2

= -2 Re (X, X) + IXl^2. Then

jx 1x1

2

j ~ (1+2K) 1x 1

2

.

167

It is not hard to see that this yields the lemma. For example, let a be an integral

curve of X and lets be an arc length parameter along a. Then wherever IXI ;::: 1,^8

we have

:s 1x 1

2

~ (1+2K)1x 1

2

.

Integrating this ODI implies the completeness of X. D

One may extend the canonical form for shrinkers, characterized in Lemma

30 .29, to the nongradient case. Let (Mn,g,X, -1) be a complete shrinker with

bounded Ricci curvature, so that

2Rc+£xg-g = 0.

Define cp (t) : M --+ M by gtcp (x, t) = -iX (cp (x, t)) and cp (x, -1) = x. By

Lemma 30.38, the 1-parameter family of diffeomorphisms cp (t) is defined for all

t E (-oo, 0). Define g (t) ~ -tcp (t)* g and X (t) ~ -icp (t)* X = -i(cp (t)-^1 )*X.

We compute

:t I g ( t) = tl g (to) - to :t I cp ( t )* g ( -1) ,

t=to 0 t=to

where gt lt=to cp (t)* g (-l) = LX(to) ( cp (to)* g (-l)). Hence

8 1

otg (t) = tg (t) + £x(t)g (t).

On the other hand, with g = g (-l) and X = X (-1), we have fort E (-oo, 0)

-2 Re (g ( t)) = cp ( t )* ( -2 Re (g))

= cp (t)* (£xg -g)

= Lip(t) x (cp (t) g) - cp (t)* g

1

= LX(t)g (t) + tg (t).

Thus (Mn, g (t), X (t), f) satisfies the shrinking gradient soliton equation for all

t E (-oo,O).

EXERCISE 30. 39. Compute gtx (t).

Regarding shrinking Ricci solitons with bounded curvature in general, we have
the following result of Naber [273]. This is another application of the monotonicity
of the reduced volume based at the singular time.

(^8) Note, as an aside, that if u (s) E M-B (O,p), where p ~ 4max{const, 1}, then by (30.121)
we have IX I (u (s)) 2 1.