264 6. ENTROPY AND NO LOCAL COLLAPSINGJKiT for all t E [-Kiti, Ki (T - ti)). Since limi-+oo JKiT = oo, goo (t) is
A;-noncollapsed on all scales from Lemma 6.48 (regarding a limit property
of sequences of A;-noncollapsed solutions). D5.3.2. Ruling out the cigar as a finite time singularity model. Finally,
we observe that Perelman's no local collapsing theorem implies that the
cigar (product with any fl.at solution like IRn-^2 or a torus rn-^2 ) cannot be a
limit of dilations about a finite time singularity as in Theorem 6.68. This is
because the cigar (product with any fl.at solution) is not A;-noncollapsed on
all scales for any A;> 0. An easy way to see this is that the cylinder 51 x IR
is a limit of the cigar. Clearly, 51 x IR (product with any fiat solution) is not
A;-noncollapsed on all scales for any A;> 0. By the property of A;-noncollapsed
being preserved under limits, this implies the same for the cigar (product
with any fiat solution).
6. Improved version of no local collapsing and diameter control
In this section we give a proof of Perelman's improvement of his no local
collapsing theorem to the case where one assumes, in the ball to be shown
to be noncollapsed, only the scalar curvature has an upper bound. We also
present the work of Topping [357] on diameter control. We end this section
with a variation on the proof of Perelman's no local collapsing theorem.
6.1. Improved version of no local collapsing. We first revisit and
revise Proposition 6.64. Let (J\/tn, g) be a closed Riemannian manifold and
r > 0. Again we shall consider the inequality (6.85) for μ (g, r^2 ) and the
test function w defined by (6.86). It is easy to see that the proof of Lemma
6.65 yields the following, where the estimate now involves Vol B (p, r /2),
which we had previously estimated in terms of Vol B (p, r) under a local
Ricci curvature lower bound assumption.
LEMMA 6.69 (c and the volume ratio). The constant c in (6.86) satisfies
the following bounds:(6 94) · VolB(p,r)^1 < - (4 7rr 2)-n/2 e -c < - VolB(p,r/2)'^1
Equivalently,
(6. 95 ) --n 2 1 og ( 4 7r ) + 1 og Vol B rn (p, r /2) < - c -< --n 2 1 og ( 4 7r ) + 1 og Vol B rn (p, r).
Using the above lemma, we obtain the following (the proof is similar to
the proof of Proposition 6.64).
PROPOSITION 6. 70 (Bounding μ by the scalar curvature and volume
ratio). The μ-invariant has the following upper bound in terms of local geo-
metric quantities. For any closed Riemannian manifold (J\/tn, g) , point