1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 5

(ii) Hence

2

Ca,A.g ( 2 ) =aVolA.g ( 2 )-2/n +4ae2Cs(M,A2g) n

sC(a,g)

if.A.2::1. D

1.2. Lower and upper bounds for the μ-invariant.
In this subsection we prove lower and upper bounds for the μ-invariant
in terms of r and certain geometric invariants of (Mn, g).

1.2.1. Upper bounds forμ.

Taking f = c to be constant in (17.5), we obtain the following elementary

upper bound for μ:

n

(17.20) μ(g, r) S 7 Ravg +log Vol (g) -

2

1og(47rr) - n,

where Ravg is the average scalar curvature of g.

On the other hand, we may choose f in terms of the minimizer of F.

The following is inequality (2) in Lemma 2.1 of [197].

LEMMA 17.3 (Upper bound forμ in terms of A, Vol, r, and n). For any

closed Riemannian manifold (Mn, g) and r > 0

1 n

(17.21) μ(g, r) Sr A (g) +~Vol (g) -

2

1og (47rr) - n.

PROOF. For any w with JM w^2 dμ = 1, by (17.7) and log (w^2 ) w^2 2:: -~,

we have

W (g, f, r) = r JM ( Rw^2 + 4 IVwl

2

)dμ-JMlog(w^2 )w^2 dμ - ~ log(47rr) - n

1 n

STY (g, w) + -Vol (g) - -log(47rr) - n,

. e 2

where g is defined by (17.2). Choosing wo ~ (47rr)-nl^4 e-f^012 to be the

minimizer of the functional g (g, w), we conclude that

1 n

μ(g,r) s W(g,fo,r) S r.A.(g) + ~ Vol(g)-

2

1og(47rr)-n.

D

When A (g) s 0, one may apply the scaling property ofμ to obtain the
following, which is Corollary 2.3 of [197].

COROLLARY 17.4 (Upper bound forμ when AS 0). If A (g) S 0, then

n 1

(17.22) μ(g,r) s logVol(g)-

2

1og(47rr)-n+ ~·