1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

234 6. ENTROPY AND NO LOCAL COLLAPSING

EXERCISE 6 .17. Let

(6.41)

so that w^2 = u. Show that

(6.42) w ( f ) = r ( T ( Rw

2

+ 4 IVwl

2

6 ) ) d

. g, '

7

JM +E (log (w^2 ) + ~ log(47rr) + n) w^2 μ

(6.43)

SOLUTION TO EXERCISE 6.17. We obtain (6.42) from substituting the
definition of w,

n

f = -2logw-

2

1og(47rr),

1

and \lw = --w\7 f

2

into (6.26).

EXERCISE 6.18 (Lower bounds for W 6 ). Let (Mn,g) be a closed Rie-

mannian manifold and let Rmin ~ infxEM R (x). Suppose that (g, f, r) satis-

fies the constraint JM ( 41fT )-n/^2 e-f dμ = 1. Show that for T > 0 the following

hold.

(1) If E > 0, then

(6.44) W 6 (g,J,r) 2: TRmin - ~ Vol(g) +E (~log(47rr) +n) > -oo.

(2) If E < 0, then

WE:(g, f, r) 2: -2C lcl + TRmin + E (~log (47rr) + n) > -oo,

where

_,_ 2r ( -2/n lcl

C~~Vol g) +2rCs(M,g)'

and Cs (M, g) is the constant in the Sobolev inequality (6.66).

Hence we conclude that when E < 0, for any A < oo, there ex-

ists a constant C (g, E, A) < oo such that

Wc(g, f, r) 2: -C (g, E, A)

for TE [A-^1 , A] and f E C^00 (M) with JM(41rr)-~ e-f dμ = 1.

SOLUTION TO EXERCISE 6.18. (1) If E > o, which corresponds to the
expanding case, then (6.44) follows from

(6.45) f ulogu dμ;:::: -~Vol (g) > -oo.

}M e

(2) If E < 0, which corresponds to the shrinking case, then the logarithmic

Sobolev inequality implies that the entropy has a lower bound (see Section 4