262 6. ENTROPY AND NO LOCAL COLLAPSING
The second part follows from that fact that if for some K, > 0 and r > 0
the metric g is K,-collapsed at the scaler, then there exists p EM such that
----"5:_K,. VolB(p,r)
rnThe proposition is proved. D
EQUIVALENT PROPOSITION. For every Riemannian manifold (M.n,9)'TE (0, oo), 01 and 02, there exists c = c (n, T, 01, 02) > 0 such that if for
some r E (0, VT] and A< oo we haveμ (9, r^2 ) ~-A, then for any p EM
with Re ~ -0 1 r-^2 in B (p, r) and R "5:. C2r-^2 in B (p, r) , we have
VolB (p, r) ~ K,rn,
where K, = c (n, T, 01, 02) e-A.
PROOF. The contrapositive is Proposition 6.64. In particular, the con-trapositive is: for every (M.n, 9) , T E (0, oo), 01 and 02, there exists a
constant C = C (n, T, 01, 02) such that if for some r E (0, VT] and K, > 0
there exists p E M with Re~ -C1r-^2 in B (p, r), R "5:. C2r-^2 in B (p, r),
and Vol B (p, r) < K,rn, then μ (9, r^2 ) < log K, + C. D
REMARK 6.66. From examining the proof of Proposition 6.64, in the
Equivalent Proposition we can replace the condition Re~ -C1r-^2 in B (p, r)
by
VolB (p, r) S 01 VolB (p, r/2)
since the only place where we used the Ricci curvature lower bound is forthe relative volume comparison. That is, there exists co =co (n, T, 01) > 0
such that
Vol B (p, r) ~ K,orn,
where K,o =co (n, T, 01) e-A.By assuming a lower bound on v (9) instead ofμ (9, r^2 ), we may remove
the lower bound of Ricci curvature assumption.PROPOSITION 6.67. For every Riemannian manifold (M.n,9), T E
(0, oo), and 01, there exists c = c (n, T, 01) > 0 such that if for some
r E (0, VT] and A < oo we have v (g) ~ -A, then for any p E M with
R "5:. C1r-^2 in B (p, r), we have
VolB (p, r) ~ K,rn,
where K, = c (n, T, 01) e-A.PROOF. If VolB (p, r) "5:. 3n VolB (p, r/2), then by the above remark,
the proposition holds. So we assume that VolB (p, r) > 3nvolB (p, r/2).
Since lims--+O+ Vo~:;~,s) = 1, there exists k EN such that VolB (p,r/2k) "5:.