1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. IMPROVED VERSION OF NLC AND DIAMETER CONTROL 269

The theorem follows from plugging in the definition of 8 (g). D

Now we can apply Theorem 6.75 to the Ricci fl.ow and obtain the fol-

lowing.

COROLLARY 6.76. Let n ~ 3 and let (Mn, g(t)), t E [0, T), be a solution

of the Riccifiow on a closed manifold with T < oo. Assume that .A(g(O)) > 0.

Then there exists C = C ( n, g ( 0)) > 0 such that

(6.100) diam(M,g(t)):::; C { R+ (t)n2

1

dμg(t)·

. JM

PROOF. Note that by the monotonicity of the .A-invariant we have

.A(g(t)) ~ .A(g(O)) > 0,

and hence the theorem is applicable. Now the corollary follows from v (g (t))

~ v (g (0)). D

6.3. A variation on the proof of no local collapsing. In this sub-

section we give a modified proof that the no local collapsing theorem follows
from entropy monotonicity using a local eigenvalue estimate. We also give
a heat equation proof of a less sharp form of the global version of this eigen-
value estimate.
6.3.1. Modified proof of no local collapsing theorem. Recall Cheng's sharp
upper bound for the first eigenvalue .\1 or'the Laplacian -~ on balls with
a lower bound on the Ricci curvature [92].

THEOREM 6.77 (Cheng, local eigenvalue comparison). Let (Mn, g) be

a complete Riemannian manifold wzth Rc(g) ~ -(n - l)g. Then for any

pEM,

(6.101)

where BlHin (1) is the open ball of radius 1 in hyperbolic space lHin of sectional

curvature -1. Here .\1 denotes the first eigenvalue of the Laplacian with the

Dirichlet boundary condition.

EXERCISE 6. 78. Suppose (Mn, g) is a complete Riemannian manifold

with Re(§) ~ (n - l)Kg, where K :S 0. Given r > 0, determine an upper

bound for .\ 1 ( B (p, r)) in terms of the corresponding model space.

We now give the modified proof of no local collapsing using the above
eigenvalue estimate and Jensen's inequality. Given the monotonicity ofμ,
the first proof we presented relies on inequality (6.83) giving an upper bound
for μ in terms of the volume ratio; it is this inequality for which we give a
second proof. Recall from ( 6.85) that we have for a Riemannian manifold