1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

92 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS

PROOF. Let H (x,p 0 , t) = (41T't)-nl^2 e-f(x,t) be the heat kernel (we drop
the subscript Po on J). One can show that

02'.W!in(f(t),t)=t f J\7HJ

2

dμ- f HlnHdμ-'!!..ln(41T't)-n.

JM H JM 2
Suppose property (1) holds. Then by Li and Yau's upper bound for
the heat kernel on a manifold with Re 2: 0 (see Corollary 3.1 in [121]
and compare with Corollary 26.26 below), there exists a universal constant

C < oo such that

H (x,po, t):::; f 0) e-d(x;ft,ol2 :::; C cn/2

VolB Po, t c

for allpo,x EM and t E (0,oo). Thus

Wrin (f (t), t) 2: - JM HlnH dμ - ~ ln (41T't) - n

C n

2: - ln-;--

2

1n ( 47!') - n

for all t E ( 0, oo). This is the desired bound for Wlin.

Conversely, suppose property (2) holds. We shall use the bound for Wlin
to estimate volume ratios. Recall that the Li and Yau differential Harnack
estimate says that

/J..H-JV'HJ

2

!!_H>O.

H + 2t -

Then

(19.15)

On the other hand, by Li and Yau's lower bound for the heat kernel on a
manifold with Re 2: 0 (see Theorem 4.1 in [121] and compare with Theorem
26.31 below), there exists a constant C (n) < oo depending only on n such
that

. C(n) _d(x,p 0 )^2
H (x,p 0 , t) 2: (

0

) e 3t

VolB po, t

for all Po, x EM. Thus

- f HlnH dμ:::; - f Hln ( C/n)

0

) e_dC"':f't^0 l

2

) dμ (x)

JM JM VolB po, t

(19.16) :::; C (n) + ln VolB (po, 0)

for some C (n) < oo, since JM Hd(xf.to)

2
dμ (x) is bounded from above by
a constant depending only on n (for this last fact, see [138] or it may be
deduced from the methods in Chapter 26 in this volume).