1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

244 6. ENTROPY AND NO LOCAL COLLAPSING

Hence, if T 2: 1, we have

(6.63)

Since >.(g) > 0, we have lim 7 -+oo μ (g, T) = +oo. D

EXERCISE 6.32. Show that if >.(g) < 0, then limT-700 μ (g, T) = -()(). In

particular, if >.(g) < 0, then v (g) = -oo.

SOLUTION TO EXERCISE 6.32. Since >.(g) < 0, there exists Jo with

JM e-f^0 dμ = 1 and

a~ JM (R + j\7 fol^2 ) e-fodμ < O.

Define J ~Jo - ~log (47rT) so that JM(47rT)-nl^2 e-f dμ = 1. We have fo~ all

T>O

μ (g,T) ~ W (g, J,T) = JM'[T ( R + /\7f/

2

) + J - n] (47rT)-nl^2 e-f dμ

~ T JM ( R +IV fol^2 ) e-fodμ +~JM (47rT)-nl^2 dμ

(47rT)-n/2

=; aT + Vol (g) ,

e

since xe-x ~ ~ for x > 0. The result follows from a< 0.

When T ---+ O+, we have

LEMMA 6.33 (Behavior of μ (g, T) for T small). Suppose (Mn, g) is a

closed Riemannian manifold.

(i) There exists f > 0 such that

μ (g, T) < 0 for all TE (0, f).

(ii)

lim μ (g, T) = 0.

T-+0+

The proof of Lemma 6.33 will be given elsewhere.

From Lemmas 6.30 and 6.33, we have

COROLLARY 6.34. If >. (g) > 0, then v (g) is ·well defined and finite.

Also, there exists T > 0 such that v (g) = μ (g, T).

3.3. Monotonicity of v and the second proof. The following lemma

is the monotonicity property of v(g(t)) along the Ricci flow g(t).

LEMMA 6.35 (v-invariant monotonicity). Let (Mn, g (t)), t E [O, T), be

a solution to the Ricci flow on a closed manifold.

(1) The invariant v(g(t)) is nondecreasing on [O, T), as long as v(g(t))

is well defined and finite..