1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

208 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS

On the other hand, since M admits a metric with positive scalar curvature,
by a result of Gromov and Lawson [125] and Schoen and Ya u [354], M does not
admit a flat metric. D

4.2. The zero case-M^3 admits a flat metric.

The borderline Case II is a bit more delicate because only asymptotically does

Rmin (t) approach zero. In particular, fort < oo, it is not clear whether the solution

g (t) behaves more like it has positive, zero, or negative curvature.

PROPOSITION 32.11. If a noncollapsed nonsingular solution (M^3 , g (t)), t E

[O, oo), to the NRF on a closed manifold satisfies limt-+oo Rmin ( t) = 0 for some to ,

then M admits a fiat metric.

We shall use three subcases to prove this. Again, let g (i) be the solution to

the Ricci flow with g (0) = g (0). If the maximal time interval [O, T) of existence

of g (i) is finite, then sup M x [o,'.t) Rmax = oo. By the proof of Proposition 32.10,

there exists a limit solution g 00 (t) on M 00 ~ M which is either flat or has positive
sectional curvature. The latter cannot happen since then Rg(t) would b e positive
for some t E [O, oo), a contradiction.
For the rest of the proof of Proposition 32.11, we shall assume that T = oo.
Recall that the evolution of the volume for the Ricci flow is

(32.13) J_Vol (g(i)) = -f (i) Vol (g(i)).

dt

This motivates dividing the analysis of Case II with T = oo into the following three
subcases:

A. (Positive) There exists a sequence ii ---+ oo such that

(32.14)

B. (Negative) There exists a sequence ii ---+ oo such that

(32.15)

C. (Zero) There exist constants c > 0 and C < oo such that

(32.16) cs; Vol(g(i))::; C

for all i E [O, oo).

Subcase A: Recall that g (ti)= 'ljJ (ii ) g (ii)· By (32.14), we have

0 +-Vol (g (ii)) = 'ljJ (ii)-

312

Vol (g (ti )) = C · 'ljJ (ii)-

312

.

Thus 'ljJ (ii) ---+ oo. By Lemma 32.8 and since Rg(t) cannot be positive, the limit
solution g 00 (t) is flat.

Subcase B: By (32.15) and (32.13), we have

foi, JM Rg(T)dμg(T)dT =-Vol (g (ii)) +Vol (g (0)) ---+ -oo.

Hence there exists a sequence of times t~ ---+ oo with

(32.17)