1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

340 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

where v denotes the unit outward normal vector field to M on 8M, where

Vx denotes the directional derivative in the direction v with respect to the

x variable, and where C1 is as in (26.18).

Since Hn (x, r; y, v) = 0 for x E 8M and Hn (x, r; y, v) 2:: 0 for x EM,

we have Vx (Hn) ::; 0. Hence

d~ JM Hn (x,r;y,v)dμg(r) (x) :S C1 JM Hn (x,r;y,v)dμg(r) (x),

which implies

JM Hn (x,r;y,v) dμg(r) (x) :S eCi(r-v).

(2) Let y E int (M) and let r be as in (26.30). Let 'ljJ: [O, oo)---+ [O, 1] be

as in (25.83) and define the cutoff function

<p : M x [O, T] ---+ ~+

by

<p ( x,r )='="''(dg(r)(x,y)). ~.

r

Note that <p is C^00 and

(26.31) supp(cp) C LJ Bg(r) (y,2r) x {r} C int(M) x [O,T].

rE[O,T]

We also have <p (x, r) = 1 for x E Bg(r) (y, r) and r E [O, T].

Using Lx,rHD = 0 and (26.3), we compute

d

d f cp(x,r)Hn (x,r;y,v)dμ 9 (r) (x)

Tj'M

(26.32) =JM <p (x, r) (f:.x,rHD) (x, r; y, v) dμg(r) (x)

+JM f (8cp or (x, r) + (R-Q) <p (x, r) ). Hn (x, r; y, · v) dμg(r) (x).

Since gr9ij = 2Rij and A= SUPMx[O,T] IRij (x,r)lg(r) < oo implies

I :r dg(r) I :S Adg(r),

we have

ocp J::i ( x, r ) ~ "',1 ~ (dg(r) (x, y)) ~~d J::i g(r) .. ( x, y .)

ur r rur

2:: -/CJcp(x,r)Adg(r) (x,y)

r

(26.33) ;:::: -2-ICA,