1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

384 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

in B ( 0, p/VK) x (0, l]. By the gradient estimate (26.142), we have

Ii (y, t)I::; C1 + }Kcn,K,1

for all t E (0, 1] and x, y EM with d (x, y) < p/VK.
By the diffeomorphism invariance of the norm of the Hessian, we have

I Hess 9 fl 9 (x, t) = I Hess gf /g (0, t).

Now we have

and

(26.148)

in iJ ( o, p/VK) x (o, 1].

Recall that a local coordinate system {xi} ~=l is called harmonic if

~(xi) = 0 for all i, i.e., gjkqk = 0.

Since !sect (9)1 ::; K and inj 9 (0) 2: p/VK, by Jost and Karcher [102]

(cf. DeTurck and Kazdan [53]), there exist Po E (0, p/VK], depending only

on n and K, and harmonic coordinates {yi}: 1 on B (0,po) such that

(26.149) ll.9ij llc1,1;2 ::; C12

and

(26.150)

for some constant C12 < oo depending only on n and K, where 9ij ~

9 ( 8 ~., 8 ~j). With respect to these coordinates, the heat equation is

(26.151)

where 191 ~<let (9ij)· Hence, by (26.149), (26.150), (26.148), and the interior

Schauder estimates (see Chapter 3 of Friedman [61] or §3 of Chapter IV on

pp. 51-61 of Lieberman [122]), we have

in B (0, Po/2) x [1/2, 1] for some constant C13 < oo depending only on n and

K.