1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

316 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

since llvllL=(PPf) < llvllL=(P2ro) < 00 independent of e. Provided we can

show that

.e-1 A 2-k

(25.45) lim sup II (ck) ::; c

C-too k=O

for some constant C < oo, we obtain from taking the lim sup as f,--+ oo of

the RHS of (25.44) that

(25.46) llvllL=(Pro) ::; C llvl1Ll(P 2 r 0 ) •

Now by (25.42) we have

(

m A 2-k m eCs ( 1+4VKro ) '.;j'c (n+3)n 4 6 n ) 2-k

limsup IT (ck) ::; lim II^0 1 n2(k+^2 )4

m-too k=O m-too k=O ro (Vol_g B_g (xo, ro))^2

(eCs(1+4VKro)'.;j'6~n~3)n Cn) 2 Cn.

(25.47) ::; 2 ) =;= c,

r 0 Vol_g B_g (xo, ro

where Cn ~ 2'.;j' I:k'=o(k+^2 )^2 -k < oo; this yields (25.45).

Finally, since (25.10) says u = eA-r v, by (25.46) we have

(25.48)

sup u::; eATo llvllL=(.P. )

P9 ( xo,To,ro,-r5) ro

::; CeATo llvllv(P 2 r 0 )

::; CeATo llullu(P 2 r 0 )'

where C < oo is given by (25.47). Tracing through the dependence of C

and A completes the proof of Theorem 25.2.

1.3. Summary of the proof of the parabolic mean value inequal-

ity.

Consider the special case of a subsolution v of the heat equation on a

fixed Riemannian manifold with Re?:: 0. Since g~ ::; ~v, for any p E [1, oo)

we have

{)

or ( vP) - ~g(T) ( vP) ::; 0.

Given a cutoff function 'ljJ with support in a compact domain D x [O, T], via

integrating by parts the above inequality against vP, we obtain for 0 ::; r 1 <

T2 ::; T