1547671870-The_Ricci_Flow__Chow

210 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

to the initial value problem

{

dr/dt = 2r^2 /n

r(to) p

By the maximum principle, Rinf (t) ~ infxEMn R (x, t ) satisfies the inequal-
ity Rinf (t) ?: r (t) for as long as both solutions exist. 0

Once we know a solution becomes singular in finite time, Theorem 6.45
tells us that its Riemannian curvatures must blow up as we approach the
singularity time.

COROLLARY 6.54. Any solution (M^3 ,g(t)) of the unnormalized Ricci

fiow on a compact manifold whose Ricci curvature is positive initially exists

on a maximal time interval 0 ::; t < T < oo and has the property that

lim (sup IRm(x,t)i) = oo.
t/T xEM3
By combining this fact with results of Sections 5 and 6, we can obtain
global pinching estimates for the curvature.

LEMMA 6.55. Let (M^3 , g (t)) be a solution of the unnormalized Ricci

fiow on a compact manifold whose Ricci curvature is positive initially. Then

the solution becomes singular at some time T < oo. Moreover, it obeys the

following a priori estimates.

(1) There exist positive constants C and 'Y depending only on the initial

data such that

~ R· > l-CR--y

R max - max

for all times 0 :'S t < T. In particular, Rmin/ Rmax--> 1 as t / T.

(2) For x E M^3 and t E [O, T), let>. (x, t) ?: μ (x, t) ?: v (x, t) denote

the eigenvalues of the curvature operator at (x, t). Then for any

E E ( 0, 1), there exists Tc. E [ 0, T) such that for all times Tc. ::; t < T,

one has

min v (x, t) ?: (1 - c) max>. (y, t) > 0.

xEM^3 yEM^3

In particular, the solution eventually attains positive sectional cur-

vature everywhere.

PROOF. By Lemma 6.53, we know the solution becomes singular at some

time T < oo. Since c [Rel ::; [Rml ::; C [Rel in dimension n = 3, it follows

from Theorem 6.45 that

(6.53) lim (sup [Re (x, t)i) = oo.

t/T xEM3

To prove claim ( 1), define

Rmin (t) ~ inf R (x, t)

xEM^3

and Rmax (t) ~ sup R (x, t).

xEM^3