1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

28 17. ENTROPY, μ,-INVARIANT, AND FINITE TIME SINGULARITIES

there exists a constant C < oo such that

(17.81) F' (r) :::; G (r) + CrF (r).
Now we proceed to estimate G (r) from above. Let <p E C 0 (B) be a
radial function (i.e., a function of d (·,p)), where B = B (p, inj (p)). We have

1 1

inj(p) 1 aw

\Jw 00 • \J<p dμ = dr

0

00

<p^1 (r) dO'

B 0 S(p,r) r

tnj(p)

=lo <p^1 (r) A (r) G (r) dr,

where

A (r) = { dO'

1 S(p,r)

is the ( n - 1 )-dimensional volume of S (p, r). Since w 00 is a weak solution
to (17.77),

(17.82)

tnj(p)

lo <p

1

(r) A (r) G (r) dr

= ~ L W 00 log (w 00 ) i.p dμ

• ~ L (-R + ~ log(47r) + n + μ(g, 1)) w 00 <pdμ.

Let

(17.83) L (r) = 2Al( ) r Woo log Woo dO'

r 1 S(p,r)

and

(17.84) K (r) =

4 A\r) ls(p,r) ( R - ~log( 47r) - n - μ(g, 1)) w^00 dO',

so that (17.82) implies that G (r) satisfies

tnj(p)

lo ('P^1 (r) G (~) - <p (r) L (r) + <p (r) K (r)) A (r) dr = O

for all radial functions <p E C 0 (B). This implies (Rothaus says, 'By the
usual one-dimensional regularity result, ... ')

(17.85)

d.

dr (G(r)A(r)) = (K(r)-L(r))A(r).

Now from definition (17.84) we have

K(r):SAC() r WoodO'

r 1 S(p,r)

=CF(r),