1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 3

The v-invariant is

(17.10) v(g) ~inf {μ(g,T): TE JR+}.

1.1.2. Monotonicity of Perelman's entropy.

Under the coupled Ricci flow system

(17.lla)

a

at9ij = -2~j,

(17.llb) af at= -b:..f + I V'/I z -R+ 2T' n

(17.llc)

dT

-=-1

dt '

we have

(17.12)

Indeed, let

v ~ [T (R+ 2b:../-IY'Jl^2 ) + f-n] u,

which satisfies W (g, f, T) =JM v dμ. We have (see Lemma 6.8 in Part I)

(17.13) O*v = -2T !Re +V'V' f - ;

7

gl

2

u,

where D* = -%t - b:.. + R is the adjoint heat operator. This implies (17.12)

since

dW = { (~ - R) v dμ = - { D*v dμ.

dt JM at JM

We remark that formula (17.13) is central to the proof of Perelman's differ-

ential Harnack estimate (see Chapter 16 in Part II).

The functional W (g, · , T) is bounded from below under the constraint

JM u dμ = 1 and there exists a smooth minimizer f n which satisfies the

equation

(17.14)

(see Proposition 17.24 below). In terms of wT = (4nT)-nl^4 e-frf^2 , this is

(17.15)

T (-4b:..wT + RwT) - wT log ( w;) - (~ log(4nT) + n) wT = μ (g, T) wT.

1.1.3. Logarithmic Sobolev inequality.
The logarithmic Sobolev inequality is intimately tied to Perelman's en-
tropy functional due to their related forms. In our discussion of this, we
shall assume n 2: 3; we leave it to the reader to verify that these results
carry over to the case n = 2 with only minor adjustments. The following is
given as Lemma 6.36 in Part I.