1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. THE POSITIVE AND ZERO CASES OF NONSINGULAR SOLUTIONS 207

STEP 2. M is diffeomorphic to a space form. Since g 00 (t) has finite volume,

M 00 must be compact (a complete noncompact Riemannian manifold with nonneg-

ative Ricci curvature must have at least linear volume growth and, in particular,

infinite volume; see Yau [446]). This implies that M 00 is diffeomorphic to M and

that r 00 (t) = r 900 (t)· Hence g 00 (t) is a solution to the NRF. Now, by the classifica-

tion of solutions to the Ricci flow on closed 3 -manifolds with nonnegative sectional

curvature (see Hamilton [136] and Shi [371]), we have that (M 00 , g 00 (t)) is either

(i) a fl.at manifold,

(ii) the compact quotient of the product of a positively curved 2-sphere and

IR, or

(iii) has positive sectional curvature. In this case, by Hamilton's theorem,

M 00 is diffeomorphic to a spherical space form.

We now rule out case (ii). Suppose that a finite cover of (M 00 ,g 00 (t)) is

isometric to ( S^2 , h 00 ( t)) x S^1 (p 00 ( t)), where h 00 ( t) is a positively curved solution

S^2 of

(32.12)

where p 00 (t) is the length of S^1 and where r 00 (t) = r 900 (t) = rh 00 (t)· Because of

the factor ~ # 1 in (32.12), we have by the Gauss-Bonnet formula that

d 1 8

• Area (h ) = --r Area (h ) = --1r.
  dt^00 3 00 00 3
  Hence h 00 (t) only exists up to a finite singular time, contradicting that g 00 (t) exists
  for all time. 0

REMARK 32.9. Alternatively, case (ii) may be ruled out by P erelman's no local

collapsing theorem applied to g 00 (t) with t ---t -oo or by the result in [153] by

Ilmanen and one of the authors.

4. The positive and zero cases of nonsingular solutions

In t his section we prove the following:
(1) Positive case: If Rmin (t) is positive after some finite time, then (M^3 ,g(t))
converges to a spherical space form under the NRF.
(2) Negative case: If Rmin (t) tends to zero as t ---t oo, then for some sequence
ti ---t oo we have that g (ti) converges to a fl.at metric on M^3.

4.1. The positive case-M^3 is a spherical space form.

For Case I , we now prove the following.

PROPOSITION 32 .10. If a noncollapsed nonsingular solution (M^3 , g (t)), t E

[O, oo), to the NRF on a closed manifold satisfies Rmin (to) > 0 for some to, then

M is diff eomorphic to a spherical space form.

PROOF. Since Rmin (to) > 0, we have T < oo. So, by the Hamilton-Ivey

estimate, there exists a sequence of times ti ---t T such that Rma:x (ti) ---t oo. Now,

since the ti satisfy

Rmax (ti) = '1/J (ti) Rmax (ti) :::; C'lj; (ti),

by Lemm a 32.8 we have that M is diffeomorphic to a space form with nonnegative

curvature.