1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. RIEMANNIAN GEOMETRY 451

As a consequence, we have the following characterization of Euclidean

space.

COROLLARY A.5 (Volume characterization of IB.n). If (Mn, g) is a com-

plete noncompact Riemannian manifold with Re 2: 0 and if for some p EM,

lim VolB (p, r) = Wn,

r->oo rn

then (M, g) is isometric to Euclidean space.

The following result about the volume growth of complete manifolds

with nonnegative Ricci curvature is due to Yau (compare with the proof of

Theorem 2.92).

COROLLARY A.6 (Re 2: 0 has at least linear volume growth). There

exists a constant c ( n) > 0 depending only on n such that if (Mn, g) is a

complete Riemannian manifold with nonnegative Ricci curvature and p E

Mn, then

VolB(p,r) 2: c(n)VolB(p,1) ·r

for any r E [1,2diam(M)).^1

The asymptotic volume ratio of a complete Riemannian manifold

(Mn, g) with Re 2: 0 is defined by

(A.7) AVR(g)~ lim VolB(p,r),

r->oo Wnrn

where Wn is the volume of the unit ball in IB.n. By the Bishop-Gromov
volume comparison theorem, AVR (g) :S 1. Again assuming Re (g) 2: 0, we
have for s 2: r,

(1)

sn-l

A (s) :SA (r) nl' r -

where A (s) ~ Vol8B (p, s),

(2)

(A.8) VolB(p,r) > A(s) ;:::AVR(g).

Wnrn nWnSn-l
We have the following relation between volume ratios and the injectivity
radius in the presence of a curvature bound (see for example Theorem 5.42
of [111]).

THEOREM A.7 (Cheeger, Gromov, and Taylor). Given c > 0, ro > 0,

and n E N, there exists io > 0 such that if (Mn, g) is a complete Riemannian

manifold with [sect[ :S 1 and if p E M is such that

VolB (p, ro)

n 2: c,

ro

(^1) We allow the noncompact case where diam (M) = oo.