1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

206 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

where the first equality is by integration by parts. Hence

J

r H h dμg(t) I 2: e -

1

~~

2

r H h dμg(t) I -2: e -

1

~~

2

M t=O JM t=t

since t :::; c^2 and since

JM H hdμg(f) = h (x, f) = 1,

where for the first equality we used that H satisfies (22.55b) and for the
second equality we used A 2: 67n. We have proved

1

H h dμ (t) I > 1 - l~c

2

> 1 - -

2

9 - b2 - 5A2' -
M t=O
which implies (22.86) as desired. This completes the proof of (22.64).

4. Logarithmic Sobolev inequality via the isoperimetric inequality

Motivated by Theorem 22.15 below, we make the following definition.

DEFINITION 22.14 (Logarithmic Sobolev inequality). We say that a Rie-

mannian manifold (Mn, g) satisfies the logarithmic Sobolev inequality

if there exists a constant Gp > -oo such that

(22.87) JM (~l\7fJ

2

+ f - n) udμ 2: Gp

for all u ~ (2?T)-n/^2 e-f such that JM udμ = 1.

Given any 'I/; E W^1 >^2 (M), define f by

'l/;2 ~ (21T)-n/2 e-f JM 'l/;2 dμ = u JM 'l/;2 dμ.

From this it is easy to see that (22.87) is equivalent to

(22.88) JM (2J\7'1/;J

2

• 'l/;

2

log'l/;

2

) dμ +log (JM 'l/;

2

dμ) JM 'l/;

2

dμ

2: (~ log(2?T) + n +Gp) JM 'l/;

2

dμ.

The following sharp logarithmic Sobolev inequality was proved for Eu-

clidean space by Gross [79] (for Euclidean space we have Gp= 0 in (22.87);

see (6.70) in Part I).

THEOREM 22.15 (Euclidean logarithmic Sobolev inequality). For any

W^1 >^2 function 'P on ~n' we have

r (2J\7cpJ^2 -cp^2 logcp^2 ) dμJR.n+log( r cp^2 dμJR_n) r cp^2 dμJR_n

~ ~ ~

(22.89) 2: (~ log(2?T) + n) ln cp^2 dμlR_n.