1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

160 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

We shall use the following elementary result.

LEMMA 30.29 (Characterization of canonical form for shrinkers). Let (Mn, g (t),

f (t), t ), t E (-oo, 0), be a complete shrinking Ricci soliton with bounde_d curvature.

Then %f (t) = IVg(t) f (t)l~(t) if and only if this soliton is in canonical form; i.e.,

(30.102) g(t)=-tcp(t)*g(-l) and f(t)=f(-l)ocp(t),

where cp (t) is defined by (30.104) below.

PROOF. By hypothes is,

8 1

(30.103) atg (t) = -2 Re g(t) = 2v^9 (tlv^9 (t) f (t) + tg (t).

(1) Suppose %f (t) = IVg(t) f (t)l~(t)' From Corollary 27.7, we may define dif-

feomorphisms cp(t) : M ~ M, t E ( -oo, 0), by

(30.104a) :tcp(x,t) = ~t (v^9 ( -^1 lf(-1)) (cp(x, t)),

(30.104b) cp (-1) = idM.

We then need to show that (30.102) holds. Define

(30.105) g(t) ~ -tcp(t)*g(-l) and f (t) ~ f (-1) ocp(t).

By Theorem 4.1 in [77] (or an easy calculation), we have

(30.106) :tg (t) = 2\7.§(tlv§(tl j (t) + ~g (t) = -2Rc 9 (t),

(30.107) of (t) = Jvg(t) J (t)J2.

8t g(t)

Since g(-1) = g(-1), by Chen and Zhu's uniqueness theorem and Kotschwar's

backwards uniqueness theorem, we have g (t) = g (t) for all t E (-oo, 0).

Thus (30.103) and (30.106) imply

vg(t)vg(t) J (t) = vg(t)vg(t) f (t).

By %f (t) = IVg(t) f (t)l~(t) and (30.107), we obtain

and

:t \7^9 (t) f (t) = \7^9 (t) ~~ (t) + 2 Rc 9 (t) ( v^9 <tl f (t))

= 2 ( v^9 <tlv^9 (t) f (t) + Rc 9 (t)) ( \7^9 (t) f (t))

= -~\lg(t) f (t)

t

:t \7^9 (t) f (t) = \7^9 (t) ~{ (t) + 2 Rcg(t) ( v^9 <t) }(t))

= -~\lg(t) j (t).

t

From this and v^9 <-^1 lf(-l) = Vg(-llj(-1), we obtain \lg(t)f(t) = \lg(t)f(t).
Thus

of (t) = Jvg(t) J (t)l 2 =of (t).

8t g(t) 8t