254 6. ENTROPY AND NO LOCAL COLLAPSING
The next exercise shows that on small enough scales 9 is K-noncollapsed
for some K.
EXERCISE 6.52. Show that for any Riemannian manifold (J\/tn,9), x E
M, and K < Wn, there exists p (x) > 0 such that for every r E (0, p (x)], we
have
and
and
IRml ~ r-^2 in B (x, r)
VolB(x, r)
---->K. rn -SOLUTION TO EXERCISE 6.52. This follows from the facts that
lim r^2 sup I Rm I = 0
r-+O B(x,r)lim VolB(x, r) = Wn.
r-+O rn
REMARK 6.53. In some sense Exercise 6.52 is a local version of Remark
6.46(2).
5.1.2. K,-noncollapsing and injectivity radius lower bound. We now show
that K-noncollapsing and a lower bound of the injectivity radius are equiv-
alent.
LEMMA 6.54. Let ( J\/tn, 9) be a complete Riemannian manifold and fix
p E (0, oo].
(i) If the metric 9 is not K-collapsed below the scale p for some K > 0,then there exists a constant IS = IS ( n, K) which is independent of
p and 9 such that for any x E M and r < p, if IRml ~ r-^2 in
B (x,r), then inj (x) 2: !Sr.(ii) Suppose that for any x EM and r < p with IRml ~ r-^2 in B (x, r)
we have inj (x) 2: !Sr for some IS> 0. Then there exists a constant
K = K (n, IS), independent of p and 9, such that 9 is not K-collapsed
below the scale p.PROOF. (i) Let B (x, r) be a ball satisfying IRml ~ r-^2 in B (x, r) for
some r ~ p. Consider the metric r-^29 on B (x, r) = Br-2§ (x, 1). Since 9 is
not K-collapsed on B (x,r), we have IRmr-2fJI ~ 1 in Br-2§ (x, 1) and
Vol9 B (x, r)Volr-2§ Br-2§ (x, 1) = n 2: K.
r
By a result of Cheeger, Gromov, and Taylor (see Theorem A.7), there exists
IS= IS (n, K) such that injr-2§ (x) 2: IS. Hence inj (x) 2: !Sr.
(ii) Again let B (x, r) be a ball satisfying IRml ~ r-^2 in B (x, r) for
some r ~ p, and consider the metric r-^29 on B (x, r) = Br-2§ (x, 1). We
have IRmr-291 ~ 1 and injr-2§ (x) 2: IS. By the Bishop-Gromov volume (or