1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 309

LEMMA 25.3.

(25.19) 1T

2

r IV' ('!j;vP)l~(T) dμg(T) dT :S L1T

2

r v^2 Pdμg(T) dT

T1 Jn T1 Jn

and

(25.20) (L 'l/;^2 v^2 Pdμ 9 (T)) (T2) :S 2L 1:

2

L v^2 Pdμ 9 (T) dT,

where supp('!/;) c D x [T1, T2] and D c n.

Inequality (25.19) is sometimes called a 'reverse Poincare-type inequal-

ity' since it is roughly of the form J IV' fl^2 :SC J f^2.

Since (25.5) implies

--n/2 -n/2

C 0 dμ9 :S dμ 9 (T) :S C 0 dμ9

and lalg :S c~/^2 lalg(T) for any 1-form a, inequality (25.19) implies (using g

instead of g ( T))

1

T2 r IV' ('!j;vP)I~ dμ_g dT :S Cont21T2 r IV' ('!j;vP)l~(T) dμg(T) dT

T1 Jn T1 Jn

(25.21)

Similarly, (25.20) implies

(25.22) (L 'lj;^2 v^2 Pdμ,q) (72) :S 2C'[JL 1:

2

L v^2 Pdμ_gdT.

STEP 2. Bounding higher LP-norms of v by lower LP-norms. To make
use of the reverse Poincare-type inequality (25.21), we now recall the follow-
ing version of the L^2 -Sobolev inequality (see [165]).

PROPOSITION 25.4 (L^2 -Sobolev inequality in a ball). There exists a con-

stant Cs< oo depending only on n such that for any complete Riemannian

manifold (Mn,9), where n 2: 3, if K 2: 0, xo EM, and r E (0, oo) are such

that

Rc,q 2:-K inB_g(xo,2r),

then

:S e^0 s(l+VKr) Vol§~ B,q (xo, r) r (r^2 1v f I~ + f^2 ) dμ_g

JB1j(xo,r)

for any 000 function f with compact support in B,q (xo, r) ..