1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

210 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS

5. The negative case-sequential limits must be hyperbolic

In this section, by just analyzing the evolution of the scalar curvature, we prove

that, in Case III, Any asymptotic limit is hyperbolic.

Recall that by Lemma 32.3 if limt--+oo Rmin (t) < 0, then Rmin (t) is a strictly

increasing function of time unless g (t) is hyperbolic. By rescaling the initial metric,

we may assume without loss of generality that

(32.19) lim Rmin (t) = -6 < 0.

t-too

The following result, although elementary in that it is a consequence of the evolution

equation for the scalar curvature, is fundamental to understanding Case III.

PROPOSITION 32.12 (Case III: Any limit is hyperbolic). Let (M^3 ,g(t)) be a

noncollapsed nonsingular solution of the NRF satisfying (32.19). Let ti ---+ oo and

let {xi} satisfy (32.6). Then any limit solution (M~,g 00 (t),x 00 ), t E (-00,00), to

the NRF corresponding to { (Xi, ti)} is a complete (either compact or noncompact)

hyperbolic manifold with Vol(g 00 (t))::::; Vol(g(t)). Moreover, for all t E JR,

(32.20) r 00 (t) = Hm rg;(t) = -6.

i-too

W e call (M~,g 00 (t),x 00 ) a hyperbolic limit of the solution (M^3 ,g(t)).

PROOF. Recall from (32.4) that

d 2

dtRmin?: -3Rmin (r - Rmin).

Since Rmin (t)::::; -6 for all t E [O, oo), we have

d

dt Rmin ?: 4 (r - Rmin) ?: 0.

Integrating this yields

(32.21) loo (r(t) - Rmin (t)) dt::::; -6 - Rmin (0) ~ C < 00.

Since the integrand r(t) - Rmin (t) is nonnegative, for every€ > 0, there exists a

time te: < oo such that for every t E [te:, oo),

i

t+l

0::::; t (r(T) - Rmin (7)) dT::::; €.

Taking the limit, we obtain for every t E JR,

(32.22) lim Rmin (t) = lim r (t) = -6,

t-too t-too

which implies (32.20).