1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE FUNCTIONALS μ AND v 235

of this chapter). In particular, by taking a = l~ in (6.65) below, we have

for£< 0,

WE:(g, f, T) 2:: JM ( 4T l'Vwl^2 + cw^2 log ( w^2 )) dμ

• TRmin +£(~log (47rT) + n)

(6.46) 2:: -2 lcl C (~:i, g) + TRmin +£(~log (47rT) + n) ~ -oo,

where

(

2T ) _ 2T -2/n 4 lcl

C ~,g - ~ Vol(g) + 2Tn 2 e 2 C 8 (M,g)

is as in Lemma 6.36.

REMARK 6.19. On a Riemannian manifold (Mn, [J) , Jensen's inequality

says that if 'P is convex on JR and u E L^1 (.All) , then

(. ) r. <po u dμ, ?. <p ( ( • ) r. u dμ).

Vol M JM Vol M JM

Applying Jensen's inequality with 'P (x) = x log x, we see that for any posi-

tive function u with JM udμ = 1,

(6.47) JM ulogu dμ 2:: -log (vol (M)).

This is an alternative estimate to (6.45).

2. The functionalsμ and v

Similarly to defining >. (g) using the energy :F (g, f) , we define two func-

tionalsμ and v using the entropy W(g, f, T). In this section we first discuss

the elementary properties of μ and v obtained directly from the properties

of the entropY. W. We show using the logarithmic Sobolev inequality that

μ(g, T) is finite and we show that a constrained minimizer f of W(g, f, T)

exists. We end this section with the monotonicity of μ.

2.1. The diffeomorphism-invariant functionalsμ and v. Let u ~

(47rT)-nl^2 e-f as in (6.3). We define a subset X of Wtet x C^00 (M) x JR+ by

(6.48) X = { (g,j,T): JM udμ = 1}.

Note that if (g, f, T) EX, then (cg, f, cT) EX for all c > O, and (g, f, T)