1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

246 6. ENTROPY AND NO LOCAL COLLAPSING

v(g(t2)). Now the theorem follows from Corollary 6.34 and Lemma 6.35(1)-

(2), which tell us that v (g (t)) is monotone and finite and characterizes when

v (g ( t)) is constant. D

4. Logarithmic Sobolev inequality

In this section we give a proof of the logarithmic Sobolev inequality

which we have used earlier. The logarithmic Sobolev inequality is related to

the usual Sobolev inequality and has the advantage of being dimensionless.

4.1. Logarithmic Sobolev inequality on manifolds.

LEMMA 6.36 (Log Sobolev inequality, version 1). Let (Mn, g) be a closed

Riemannian manifold. For any a > 0, there exists a constant C (a, g) (given

by (6.67)) such that if cp > 0 satisfies JM cp^2 dμ = 1, then

(6.65) JM 1.p2logcpdμ::; a JM IVcpl^2 dμ + C (a,g).

PROOF. Recall that the Sobolev inequality (see Lemma 2 in [245]) that

if JM cp^2 dμ = 1, then (assume n > 2)

n-2

(6.66) JM IVcpl^2 dμ 2: Cs (M,g) [JM cpn

2

~2dμ]--;;;;--v-^2 /n,

where V = Volg (M). Note that for Cn = ~e we have Cn log x::; x^2 fn for all

x > 0, so that

Cn r cp2 log cpdμ ::; r cp2+ ~ dμ ::; c r cp2+~ dμ +! r cp2 dμ,

JM JM JM c JM

for any c > 0, since cpl+~cp::; ccp^2 (1+~) + icp^2. By HOlder's inequality,

n-2 2

JM cp2cp~dμ::; (JM cp,?~2dμ )----;;;;-(JM cp2dμ);;;.

Hence, using JM cp^2 dμ = 1, we have

Cn JM cp2 log cpdμ ::; c (JM cpn2~2 dμ) n;2. + ~

::; Cs (~,g) (JM IVcpl2 dμ + v-2/n) + ~·

Inequality (6.65) follows by choosing

(6.67) C (a, g) = av-2/n + 2 2c\M )"
an e s ,g
Now we have proved the lemma when n > 2. We leave then= 2 case as
an exercise. D

EXERCISE 6.37. Prove the above lemma when n = 2.

Making the substitution cp = e-¢1^2 in (6.65), we have the following.