1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. NO FINITE TIME LOCAL COLLAPSING 259

Riemannian manifold, p EM and r E (0, p] are such that Re 2': -c 1 (n) r-^2

and RS c1 (n) r-^2 in B (p, r), then

(6.83) μ (g, A r^2 ) S log VolB n (p, r) + C2 ( n, p).

r

That is,

In particular, if for some r;, > 0 and r E (0, p] the metric 9 is r;,-collapsed at

the scale r, then

μ(9,r^2 ) :S:logr;,+C2(n,p).

Proof. As in ( 6 .41) with r = r^2 , define the positive function w by

(6.84)

From the definition (6.56) ofμ as an infimum of W, we have by rewriting

W (9, f, r^2 ) in terms of w (compare to (6.42)),

(6.85) μ (9, r^2 ) S JM r^2 ( 4 l'Vwl^2 + Rw^2 ) dμ +JM (f - n) w^2 dμ

~ K (9,w,r^2 ),

where JM w^2 dμ = 1 and f = -2 log w-~log ( 47rr^2 ). While making the con-

vention that f (y) w^2 (y) = 0 when w (y) = 0, we claim that (6.85) holds for

nonnegative Lipschitz functions w satisfying JM w^2 dμ = 1. To see this claim,
first by (6.56) we know that (6.85) holds for positive Lipschitz functions w
satisfying JM w^2 dμ = 1. Now given any nonnegative Lipschitz function w
satisfying JM w^2 dμ = 1, define for EE (0, 1),

We; ~ Cc; ( w + E) '

where the constant Cc: is defined by JM w;dμ = 1. Clearly limc:-+oCc: = 1,
We: is a positive Lipschitz function, and hence μ (9, r^2 ) :S: K (9, We:, r^2 ) for
each E E (0, 1). Using lim c:-+odog E = 0 and fc: = -2 log We: - ~log ( 47rr^2 ) in

the definition of K (9, we:, r^2 ), we have limc:-+oK (9, We:, r^2 ) = K (9, w, r^2 ).

The claim is proved.

Now let</>: [O, oo)-+ [O, 1] be a standard cut-off function with</>= 1 on

[O, 1/2], </> = 0 on [1,oo), and l</>'I :S: 3. Assumep EM and r E (O,p] are such

that we have the curvature bounds Re 2': -c1 ( n) r-^2 and R :S: ci ( n) r-^2 in

B (p, r) for c 1 (n) = n (n - 1). We make a judicious choice for f so that the

RHS of (6.85) reflects the local geometry at p with respect to the metric 9.

In particular, let

(6.86)