1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

152 20. COMPACTNESS OF THE SPACE OF 11;-SOLUTIONS

PROBLEM 20.22. Can one show that for any complete noncompact 11,-

solution with Harnack we have

lim inf R (x, t) dg(t) (x, p )^2 = oo

dg(t) (x,p )--+=

for all t E ( -oo, OJ?

Nate that a steady gradient Ricci soliton (Mn, g ( t) , f ( t)) with Re ?.

0 automatically satisfies the trace Harnack estimate since for any tangent

vector X we have

8R

at+ 2 (V7R,X) + 2Rc(X,X) = 2Rc(X + \i'f,X + \i'f) 2: 0.

The following calculations occur at t = 0. Therefore, by Theorem 20.17, on

a nonflat steady gradient Ricci soliton with Rm?. 0 which is A;-noncollapsed

at all scales, we have

(20.52) Re (V7 f, V7 f) = t (V7 R, V7 f) = t ~~ ::; ~ R^2.

Let a: [O, oo)-+ M be an integral curve to -V7 f, i.e., 0-(u) = - (V7 f) (a (u)).

Then

d~ R-^1 (a (u)) = (-V7 f, V7 (R-^1 )) = R-^2 (V7 R, V7 f) ::; 77.

This implies

COROLLARY 20.23. For a nonfiat steady gradient Ricci soliton with Rm?.

0 which is A;-noncollapsed at all scales we have

1

R (a (u)) 2: R-1 (a (0)) + 7JU.

If n = 3, then one can show from this that

R (x) > c

• 1 + d (x,p)

for some positive constant c.
Furthermore, if one invokes Perelman's 'canonical neighborhood' theo-
rem for 3-dimensional A;-solutions (see §11 of [152]), then one may obtain

the corresponding upper bound for R. For every E > 0 there exists a com-

pact set Kr; in M^3 such that every point x E M - Kr; is the center of an

s-neck. Then for any 6 > 0, there exists E > 0 such that

ll'.lRI ::; 8R^2 and 1Rcl^2 2:

1

;

8

R^2

in M-Kr; (in the model (round) cylinder case we have LlR = 0 and IRc/^2 =
~R^2 ). Then
(V7 R, V7 f) = LlR + 2 IRc/^2 2: (1 - 28) R^2.

This implies that if a : [O, oo) -+ M - Kr; is an integral curve to -\i7 f, then