252 6. ENTROPY AND NO LOCAL COLLAPSINGRicci flow with surgery may lead to the resolution of Thurston's geometriza-
tion conjecture. Perelman's deeper analysis of 3-dimensional singularity for-
mation greatly strengthens this expectation.^155.1. 11;-noncollapsing and injectivity radius lower bound.
5.1.1. K-noncollapsing on a Riemannian manifold. Let (.Mn,9) be a
complete Riemannian manifold.DEFINITION 6.44 (11;-noncollapsed). Given p E (0, oo] and 11; > 0, we say
that the metric g is 11;-noncollapsed below the scale p if for any metric
ball B(x, r) with r < p satisfying I Rm(y)J :::; r-^2 for ally E B(x, r), we have
( 6. 78 ) Vol!ix, r) ~ K.If g is K-noncollapsed below the scale oo, we say that g is K-noncollapsed
at all scales.
Complementarily, we give the following.DEFINITION 6.45 (11;-collapsed). We say that g is 11;-collapsed at the
scaler at the point x if I Rm(y)J :::; r-^2 for ally E B(x, r) and
( 6. 79 ) VolB(x,r) < K.
rnThe metric g is said to be 11;-collapsed at the scale r if there exists x E M
such that g is K-collapsed at the scaler at the point x.
Thus g is not K-non collapsed below the scale p if and only if there exists
r < p and x E M such that g is 11;-collapsed at the scale r at the point x.
REMARK 6.46.
(1) If _Mn is closed and flat, then g cannot be 11;-noncollapsed at all
scales since I Rm I = 0 :::; r-^2 for all r ·and Vol B( x, r) :::; Vol (.M)
so that limr--+oo Vol~~x,r) = 0 for all x E M.
( 2) If (_Mn, g) is a closed Riemannian manifold, then for any p > 0
there exists 11; > 0 such that g is K-noncollapsed below the scale p.We have the following elementary scaling property for 11;-noncollapsed
metrics.
LEMMA 6.47 (Scaling property of K-noncollapsed). If a metric g is K-
noncollapsed below the scale p, then for any a > 0 the metric a^2 g is K-
noncollapsed below the scale ap.
(^15) Some ;i,spects of Perelman's singularity theory are discussed in Chapter 8 of Part I
and also in Part II of this volume.