1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

5. LOGARITHMIC SOBOLEV INEQUALITY 27

REMARK 27.47. In comparison, Corollary 6.38 in Part I is equivalent to the

following statement. Given a ny closed Riemannian manifold (Mn, g) and constant

b > 0, there exists a constant C (b, g) such that if a function cp satisfies JM e-'Pdμ =

( 471" t^12 , then

JM (b IVcpl

2

• cp - n)e-'Pdμ;::: -C (b, g).

Here we give a heuristic proof of the theorem following the Bakry-Emery

method, which can be made rigorous in our noncompact setting; detailed proofs

are given in [53], [16], and [425]. Let 01 ~ gt -6.1 denote the f-heat operator.

Let ~(x, t) be a solution to

(27.119) o 1 ~ = IV~l^2 , equivalently, 01 es = 0.

Henceforth we shall assume suitable growth conditions on ~ so that we can

differentiate under the integral sign and so that we can integrate by parts to justify

some of the equalities below. We first observe that

(27.120) !!:_ ( es-! dμ = ( (~es) e-f dμ = ( (6.1es)e-f dμ = 0.

dtJM JM ot JM

Now assume that~ satisfies the normalization (47rt/^2 =JM es-! dμ.

Define the relative Fisher information functional

(27.121) I(~)~ JM IV~l^2 es-! dμ.

By (27.96), for any function u(x, t) we have the parabolic Bochner formula

(27.122) ~OJ 1Vul

2

= - l\7^2 ul

2

+(Vu, \7D1u) - Rei ('Vu, 'Vu).

Hence, for ~ satisfying (27.119) on a shrinking GRS, we have

(27.123) 01 l\7~1

2

= -2 l\7^2 ~1

2

+ 2(\7~, \7 l\7~1

2

) - l\7~1

2

.

From this we compute that

(27.124) D 1 (1V~l

2

es)= - es(2 IV^2 ~1

2

+ IV~l

2

).

Therefore

(27.125)! I(~ (t)) =JM :t (l\7~1

2

es)e-f dμ

= - JM es(21v2~l2 + IV~l2)e-f dμ

::;-I(~(t)).

Now define the Boltzmann relative entropy (or Nash entropy) functional

(27.126) H (~)~JM ~es-! dμ.

We compute using (27.119) that^4
(27.127)

:t H(~ (t)) =JM ( :t es ) (~ + 1) e-f dμ =JM 6. 1 (es) (~ + 1) e-f dμ = -I (~(t)).

(^4) If f = 0, then€~ In u gives H (€)=JM uln udμ and I(()= JM 1v: 12 dμ. As a special case
of (27.127), we h ave that ~ H (In u) = -I (ln u) provided u > 0 satisfies the heat equation.