1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

146 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

Before we prove Theorem 30.13 at the end of this section, we first discuss a

more precise upper bound for the reduced distance f. of Type I solutions due to

Naber [273] (compare with the more elementary Lemma 7.13(iii) in Part I, which

bounds f. in terms of the maximum curvature).

LEMMA 30.15 (The reduced distance of a Type I solution has at most quadratic
growth in space). Suppose that (Mn,g(t)), t E [a,w], where -oo <a< w < oo,
is a complete solution of the Ricci flow with bounded curvature and satisfying the
Type I condition:

(30.45)

M

1Rm 9 (t)I:::; -- on M x [a,w).

w-t

Then, for any p, q E M and€ E (0, w;",,), the reduced distance £ ~ fp,w satisfies

(30.46)

Bd^2 <) (q,x) A

£ (x, t) :::; ( t ) + B + 2£ (q, t)

2 w-t

for all x E M and t E [a+ E, w), where A < oo and B < oo are constants depending

only on n, E, and M. In particular,

Bd^2 () (p, x) A

£ (x t) <^9 t + - + 2n^2 M.

' - 2 (w -t) B

(30.47)

PROOF. Let (x,t) EM x [a+e,w), let q EM, and let (3: [0,dg(t)(q,x)]-+ M

be a minimal geodesic from q to x with respect tog (t). Define

(30.48) u ( s) ~ £ ((3 ( s) ' t).

The idea of the proof is to establish, via a gradient estimate, a first-order ODI for
u (s) to control its growth.
Let 1: [t,w] -+ M be a minimal .C-geodesic with I (t) = z and I (w) = p. By
(7.89) in Part I,

(30.49) l\7£1^2 (z, t) = -R (z , t) - K (r) + £ (z, t),

(w -t)^312 w - t

where
(30.50)

K (r) ~iw(w -i) ~ (~R -2\i'R · 1' (i) + 2Rc (1' (i) ,1' (i)) - ~)di.

t ut w - t

STEP 1. Bounding K (r). First suppose the weaker Type A assumption that

1Rm 9 (t) I :::; (w'!w on M x [a, w). By (30.3), we have on M x [a+ E , w) that

l

f)RI( ') Cnc:M

8t y' t :::; ( w ~ f) 2r and IV' RI (y, i) :::; Cn,c:,M3 ·

(w-i)^2 r