1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE FUNCTIONALS μ AND v 237

LEMMA 6.22 (Euler-Lagrange for minimizer). The Euler-Lagrange equa-

tion of (6.49) is

(6.51) T ( 2b..f - IV f 12 + R) + f - n = C.

For the minimizer f 7 of (6.49),

(6.52) T ( 2b..fT - IV fTl^2 + R) + fT - n = μ (g, T).

Compare (6.52) to the equation 2b..f - IV fl^2 + R = ,\ (g) for the mini-

mizer f of F (g, ·).

In terms of w ~ (41T'T)-nl^4 e-f 12 as in (6.41), a simple computation shows

that μ is the lowest eigenvalue of the nonlinear operator:
(6.53)
N (w) ~ -4Tb..w +TRW - (~ log(41T'T) + n) w - 2w log w = μ (g, T) w.

2.2. The finiteness ofμ and the existence of a minimizer f. As
a consequence of Exercise 6.18 we have the following.

LEMMA 6.23 (Finiteness of μ). For any given g and T > 0 on a closed

manifold Mn,

(6.54) μ(g,T) > -00

is finite.

PROOF. Since μ(Tg,T) = μ(g,1), we may assume without loss of gen-
erality that T = 1. We need to show that for any metric g there exists a
constant c = c(g) such that

(6.55) W (g, f, 1) =JM (R +IV fl^2 + f - n) (41T')-nf^2 e-f dμ 2:: c

for any smooth function f on M satisfying (41T')-n/^2 JM e-f dμ = 1. As in

(6.41) with T = 1, let w = (41T')-n/^4 e-f/^2. By (6.42), the lemma is equivalent
to showing that

W (g, f, 1) =JM ( 4 IVwl^2 + ( R-2logw - ~log (41T') - n) w^2 ) dμ

~ 'H (g' w) 2:: c

for any w > 0 such that JM w^2 dμ = 1.

Since R - n 2:: infxEM R (x) - n > -oo, it suffices to show that there

exists C < oo such that

JM w^2 logwdμ :S 2JM1Vwl

2

dμ + C

for all w > 0 with JM w^2 dμ = 1. This follows from the logarithmic Sobolev
inequality (6.65),^11 which we state in the next subsection. D

(^11) With a= 2.