1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

248 6. ENTROPY AND NO LOCAL COLLAPSING

and \7 f ~ x - 2 "'¢¢. We compute

Ln (~ l\7 Jl2 + f - n) (27r)~n/2 e-f dx

= 2 Ln ( ~ lxl

2

¢^2 - ¢ x · \7 ¢ + I \7 ¢1

2

• ¢^2 log¢ - ~¢

2

) dv,

where dv = (27r)-n/^2 e-lxl^212 dx. Now integrating by parts yields

-¢ x · \7¢dv = - n<;b dv - - lxl ¢ dv,

1

11 2 11 2 2

m;_n 2 m;_n 2 m;_n

so that we have the identity

1 (

-¢ x · \7 ¢ - n -¢^2 + -^1 lxl^2 ¢ 2) dv = 0.

m;_n 2 2

Hence

(6.72)

Ln (~1'Vfl^2 +f-n) (27r)-n^12 e-fdx=2 Ln (1\7¢1

2

-¢^2 log¢)dv,

with the constraint

Since log (Jm;.n ¢^2 dv) = 0, by (6.72) and Gross's logarithmic Sobolev inequal-

ity, we have

D

EXERCISE 6.42. Show that Gross's logarithmic Sobolev inequality for
Euclidean space implies that Euclidean space (~n, gm;) has nonnegative en-
tropy:

W (gm;, f, T) 2: 0 and μ(gm;, T) = 0.

Now we give Beckner and Pearson's proof of Gross's logarithmic Sobolev
inequality, which is a consequence of the following [23].

PROPOSITION 6.43. If fm;_n 'ljJ (x)^2 dx = 1, then

(6.73) '!!.

4

1og (--2__ f l'V'l/J (x)l^2 dx) 2: f (log l'l/J (x)I) 'l/J (x)^2 dx.

Ken }m;_n }ffi!.n

Note that this inequality is scale-invariant.

We first show that (6.73) implies Gross's logarithmic Sobolev inequality.