1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

168 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

THEOREM 30.40 (Shrinkers with IRml ~ C must be gradient and K-non-
collapsed). Let (Mn, g (t), X (t), t), t E (-oo, 0), be a complete noncompact shrink-
ing Ricci soliton with bounded curvature and in canonical form, so that
1
2Rij + \liXj + 'iljXi + tgij = 0.

Then:

(1) There exists a function f : M ---+ IR such that (M, g (t), f (t), t) is a

complet e shrinking GRS (with f normalized).^9

(2) There exists K > 0 depending only on n and J e - f dμ such that (M, g) is

K-noncollapsed below all scales.

PROOF. (1) Let g ~ g (-1), fix 0 EM, and let r (x) ~ d 9 (x, 0). By Lemma

30.38, the vector field X ~ X (-1) is complete. Let cp (t) : M ---+ M, t E (-oo, 0),

be the 1-parameter family of diffeomorphisms generated by -iX with cp (-1) =id.

By Lemma 30.37, there exists p < oo such that X satisfies

(30.122) ('ilr ,X)>l inM- B 9 (0 , p).

Hence, fort E (-oo, 0),

d / dcp ) 1 1

dt r (cp (t)) = \ \lr, dt (t) = -t (\lr, X ) (cp (t)) > -t

whenever cp(t) EM - B 9 (O,p). Hence, for any y EM, there exists t 0 < 0 such

that

cp (y, t) E B 9 (0, p) for all t ~ta.

Similarly to the proof of Proposition 30. 32 , we conclude that for any Ti-/' oo,

the sequence of pointed solutions (M, g; (t), (y, -1)) subconverges to (M, g (t),

(z,- 1)) for some z E B(O,p) (with the property that X(z) = 0). Moreover , by
the proof of Theorem 30.31 (where we do not need to assume K-noncollapsed since
we have convergence), a reduced dist ance e based at the singular time 0 yields a
gradient shrinker structure for g ( t).

(2) Let x o E M, and let Ti-/' oo. By Proposition 30.32, there exists Po E M
such that (M, gi (t), (xo, - 1)) subconverges to (M, g (t), (po, -1)). By Theorem
30.31, we have that (M, g (t), £~ 0 , 0 (t), t) is a GRS with normalized potential func-

tion being f (t) = £~ 0 , 0 (t), the reduced distance based at the singular time. Hence

Vio,o (t) ~ (-47rt)-~ JM e-e~o.o(t)dμg(t)

= (-47rt)-~ JM e-f(t)dμg(t)

= const.

By P erelman's weakened no local coll apsing theorem (see Theorem 8.26 in Part I

for an exposition), there exists K > 0 depending only on n and J e -f dμ such that

(M,g) is K-noncollapsed below all scales. 0

(^9) Note that this implies that X - \J f is a Killing vector field.