1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

270 6. ENTROPY AND NO LOCAL COLLAPSING

μ(9, r^2 ) S JM r^2 (4l'Vwl^2 + Rw^2 ) dμ

• JM (log ( w^2 ) + i log( 47rr^2 ) + n) w^2 dμ

for all w with JM w^2 dμ = 1. Using (6.47), we have for any w with supp (w) C

B(p,r) and JMw^2 dμ = 1 that

• JMlog(w^2 )w^2 dμ:::; logVolB(p,r).

By assumptions Re (9) 2:': - (n - 1) r-^2 and R:::; n (n - 1) r-^2 in B (p, r) ,^19

we have for any w with supp (w) C B(p, r) and JM w^2 dμ = 1,

JM Rw^2 dμ S n ( n - 1).

Let g = r-^2 9; then B 9 (p, 1) = B(p, r) ~ B9(p, r). By Theorem 6.77 and the

Rayleigh principle for eigenvalues,

J A l'Vwl2 dμ-

inf J _ 2 d g =Al (B9(p, 1)) S .A1(BHn(l)).

supp(w)CB9(p,l) MW μ9

Hence we have

i~f f, r^2 4l'Vwl^2 dμ S 4.A1(BHn(l)).

f;Viw dμ,=1, }M

supp(w)CB(p,r)

Therefore

log Vol~~p,r) 2:': μ(9,r^2 )-4.A1(BHn(l)) -n(n-2) + ilog(47r).

This provides the needed estimate to replace (6.83).

6.3.2. A heat equation proof of a global eigenvalue estimate. We now

recall a global version of Cheng's Theorem 6. 77.

THEOREM 6.79 (Cheng's eigenvalue estimate, global). If (.Mn,9) is a

complete noncompact Riemannian manifold with Re (9) 2:': -(n - 1)9, then

.A (-~) < (n - 1)2

1 - 4

It turns out that a weaker version of this estimate, i.e., .\1 :::; n(n 4 -l),
can be proved using the energy/ entropy computation of Perelman for the
fixed metric case; we give this proof below. First we state a formula which
is implicit in [283].

(^19) We choose these constants for our curvature bounds since they are implied by
-r-^2 ::;; sect::;; r-^2.