1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. THE THREE CASES OF NONSINGULAR SOLUTIONS 203

Now, the closure C^3 of the complement of the union of all the immortal almost

hyperbolic pieces in M is a compact 3 -manifold with boundary consisting of a

disjoint union of tori, each with area A , such that the maximum injectivity radius

of ( C, g ( t) le) can be made as small as we like by taking A sufficiently small and t

sufficiently large. By Theorem 31.55, C is a graph manifold.

3. The three cases of nonsingular solutions
   In this section we begin by observing that the noncollapsed nonsingular as-
   sumption allows us to take limits of the solution as time tends to infinity. We then
   divide the study of these limits according to whether the minimum scalar curvature
   is positive at some finite time, tends to zero, or tends to a negative constant.

3.1. Evolution of R under the NRF.

Let (Mn,g(t)), t E [O, T), be a solution to the NRF on a closed manifold.

Recall that the evolution of the scalar curvature of g (t) is given by

(32.3)

where r is the average scalar curvature. Applying the weak maximum principle to

equation (32.3) yields

(32.4) dtRmin d- (t) =inf { 8t aR (x, t): R (x, t) = Rmin (t) }

2

2: -Rmin ( t) ( Rmin ( t) - r ( t))

n

for the lower Dini derivative of the Lipschitz function Rmin (t). Consequently, we

have the following.

LEMMA 32.3 (Monotonicity of Rmin(t)). Let (Mn,g(t)) be a solution to the

NRF on a closed manifold.

(1) If Rmin (to)> 0 for some to, then Rmin (t) > 0 for all t 2 to.

( 2) If Rm in ( t) ::; 0 for all t, then Rm in ( t) is nondecreasing. If, in addition,

g ( t) is not Einstein, then Rm in ( t) is strictly increasing.

PROOF. (1) At any time where Rmin > 0, we have

d 2 2

-d lnRmin 2 - (Rmin - r) > --r.

t n n

This shows that if Rmin > 0 holds at some time, then it holds at all later times.

(2) The first statement in this part follows from (32.4) since in general Rmin (t)-

r (t) ::; 0.

To prove the second statement, suppose Rmin (ti) = Rmin (t2) ~ c ::; 0 for

some ti < t 2. Then Rmin (t) = c fort E [t1, t2]· By applying the strong maximum

principle to equation (32.3), we conclude that R (t) = c and that g (t) is Einstein.

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