1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. HEAT KERNELS UNDER CHEEGER-GROMOV LIMITS 195

EXERCISE 22.10. Use method (i), i.e., mean value inequality, to prove

(22.33).

Proof of (2). We prove a general statement, which we apply to Hi at

the very end. Let (Mn, g ( 7)), 7 E [O, w], be a complete solution to the

backward Ricci flow. Suppose that xo E M is such that

Rcg( 7 ) 2: (n - 1) K and Rg( 7 ) :::::; L in Bg(o) (xo, R)

for 7 E [O, w], where K :::::; 0, L 2: 0, and where R > 0 is chosen sufficiently

small so that Bg(O) (x 0 , R) is regular. Let

HR : Bg(O) (xo, R) x [O, w] -+ lR+

be the adjoint Dirichlet heat kernel (for its existence, see §5 of Chapter 24),

i.e.,

Then

(22.37)

D* HR ~ ( :7 -b..g(T) + Rg(T)) HR = 0,

HRlaBg(o)(xo,R) = 0 for 7 E (0, w],

lim HR ( ·, 7) = 8x 0 •

T\,0

( :

7

- b..g(T)) ( eL^7 HR) 2: eLTD* HR = 0,

lim (eL^7 HR) ( ·, 7) = 8xo·

T\,0

For comparison, let

H1;<' R' : B ( xo, R') x [O, oo) -+JR.+

'

be the Dirichlet heat kernel centered at xo for the (static) simply-connected

space form M]k, of constant sectional curvature K'. Here K' is to be chosen

below and when K' > O, we shall choose at least R' < 7r/VJ(i. Note that

H~<' R' ( · , 7) is rotationally symmetric about xo and we shall also write it as
a fu~ction of the distance to xo.
Let r = dg(T) (y, xo), let HK (r) denote the mean curvature of the dis-
tance sphere SK (xo, r) in M]k, and let H (y) denote the mean curvature at
y of the sphere S ( xo, r) in M with respect to g ( 7). Transplanting HK, R'

to M, we have '

(: 7 - b..g( 7 )) HK',R' (dg( 7 ) (y,xo) ,7)

(22.38) = (HK1(r)-H(y)+ : 7 dg( 7 )(y,xo)) (:rHK',R') (r,7)

:::::; (HK1(r)-HK(r)+(n-l)Kr) (:rHK',R') (r,7)