1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

450 A. BASIC RICCI FLOW THEORY

where the dot denotes the metric inner product, i.e., X · Y = (X, Y) =

gijxiyj.

If gtgij = -2Rij on M x (a,w) and f: M x (a,w) --t lR, then

( ~ -:t) l\7 fl

2

= 2 l\7\7 fl

2

+ 2\7 f. \7 ( ( Ll - :t) f).

1.9. The cylinder-to-ball rule. The following is an obvious modifi-

cation of Lemma 2.10 on p. 29 of Volume One.

LEMMA A.2 (Cylinder-to-ball rule). Let 0 < L ::::; oo and let g be a

warped-product metric on the topological cylinder (0, L) x sn of the form

g = dr^2 + W (r)^2 gcan,

where w : (0, L) --t lR+ and 9can is the canonical round metric of radius 1

on sn. Then g extends to a smooth metric on B ( 5, L) (as r --t O+) if and

only if

(Vl-2.16) lim w(r) = 0,

r--+0+

(Vl-2.17) lim w' (r) = 1,

T--+0+

and

(Vl-2.18)

d2kw

r--+0+ lim d r^2 k (r) =^0 for all k EN.

1.10. Volume comparison. We recall the Bishop-Gromov volume

comparison (BGVC) theorem.

THEOREM A.3 (Bishop-Gromov volume comparison). Let (Mn,g) be a

complete Riemannian manifold with Re 2: ( n - 1) K, where K E IR. Then

for any p EM,

VolB(p, r)

VolK B(pK, r)

is a nonincreasing function of r, where PK is a point in the n-dimensional

simply-connected space form of constant curvature K and VolK denotes the

volume in the space form. In particular

(A.6)

for all r > 0. Given p and r > 0, equality holds in (A.6) if and only if B(p, r)

is isometric to B (p K, r).

If Re 2: 0, we then have the following.

COROLLARY A.4 (BGVC for Re 2: 0). If (Mn, g) is a complete Rie-

mannian manifold with Re 2: 0, then for any p E M, the volume ratio

VoI~Jp,r) is a nonincreasing function of r. We have VoI~Jp,r) ::::; Wn for all

r > O, where Wn is the volume of the Euclidean unit n-ball. Equality holds

if and only if (Mn, g) is isometric to Euclidean space.