1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

258 6. ENTROPY AND NO LOCAL COLLAPSING

we have injg(t) (p) 2: er.

Corollary 6.62, i.e., Perelman's no local collapsing theorem, implies that

if (Mn, g (t)), t E [O, T), is a solution of the Ricci flow on a closed manifold

with T < oo, then g(t), t E [O, T), satisfies a local injectivity radius estimate.

5.2.2. Proof of No Local Collapsing Theorem 6.58. The idea of the proof

is that if a metric g is K-collapsed at a point x at a distance scale r for K

small and r bounded, then W (g, f, r^2 ) is negative and large in magnitude,

e.g., on the order of log K, for f concentrated in a ball of radius r centered

at x. This contradicts the monotonicity formula for μ (g ( t) , r ( t)).

PROOF OF THEOREM 6.58 ASSUMING PROPOSITION 6.64. We shall say

that r is the (space) scale of μ (g, r^2 ); the justification for this terminology

occurs below. Since T /2 < T, by the remarks after Definition 6.55, there

exists Ko = 11);0 ( n, g ( 0) , T, p) > 0 such that g ( t) is Ko-noncollapsed below the

scale p for all t E [O, T /2].

On the other hand, if t E [T /2, T), then for any 0 < r :::; p, we have

t + r^2 E [T/2, T + p^2 ), and by the monotonicity formula (6.61), we have

μ (g (t), r^2 ) 2: μ (g (0), t + r^2 )

(6.82) 2: inf μ(g(O),r)~-Ci(n,g(O),T,p)>-oo

TE[T/2,T+p2]

since T < oo. In summary, by the monotonicity formula, since the μ-

invariant of the initial metric is bounded from below at scales bounded from

above and below, the μ-invariant of the solution after a certain amount of

time (say T /2) is bounded from below at all bounded scales. The theorem

will follow from the important observation that if a Riemannian metric is

K-collapsed at some scale r for K small, then its μ-invariant is negative and

large in magnitude at the time scale r^2 (see Proposition 6.64 below).

If x EM, t E [T/2,T), and r E (O,p] are such that /Rmg(t)/ :::; 1 in

Bg(t) (x,r), then Rcg(t) 2: -c1 (n)r-^2 and Rg(t):::; ci (n)r-^2 in Bg(t) (x,r),

where c1 (n) = n (n - 1). So by (6.82) and (6.83),

-Ci (n,g (0) ,T,p):::; μ (g (t) ,r):::;^2 log Volg(t) Bg(t) n (x, r) + C2 (n,p).

r

We conclude that

Volg(t) Bg(t) (x, r) > ( (O) T ) > 0

r n - K1 n, g ' 'P '

where K1 (n, g (0), T, p) = e-Ci(n,g(O),T,p)-C^2 (n,p). The theorem follows with
the choice K(n, g(O), T, p) ~ min{Ko, K1}. D

Now we turn to bounding the μ-invariant from above by volume ratios
in a Riemannian manifold.

PROPOSITION 6.64 (μ controls volume ratios). Let p E (0, oo). There

exists a constant C2 = C2 ( n, p) < oo such that if (Mn, g) is a closed