1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

228 23. HEAT KERNEL FOR STATIC METRICS

(Minj(g) - Minj(g)/s) X lRf,

. -(n/2) ( d^2 (x,y)) ( (inj(g)/8)

2

)

JHNI (x, t, y, u) ~ C (t - u) exp - 5 (t _ u) exp - 20 (t _ u)

( ) ( )M-(n/2) ( d

2

23.45 ~const· t-u exp - (x,y))

5 (t-u)

for any M E N, where the dependence of const < oo includes M and

T. Similarly, we have estimates of the same form (with different con-
stants) for all the higher covariant derivatives and time derivatives of HN
in (Minj(g) - Minj(g)/s) x lR.f. For example, for any MEN and TE (0, oo)

there exists const < oo such that

N.

= E (x, t; y, u) "( ~ \1 x<f>k(x, y) - (d\lxd)(x,y) ) k

k=O^2 (t _ u) </>k (x, y) (t - u)

~ C (t - u)-n/2 exp (-d2 (x, y)) (i + _1_)

4(t-u) t-u

~ const · (t - u)M-(n/^2 ) exp (-d

2

(x, y))

5 (t - u)

for (x, y, t, u) E (Minj(g) - Minj(g)/s) XlR.f; similarly we may estimate J8tHNI·

An easy induction leads to

/af\l~HN/ (x, y, t, u) ~ const · (t-u)M-(n/^2 ) exp (-:~t(~ ~)

for (x, y, t, u) E (Minj(g) - Minj(g)/s) XlR.f, where the dependence of const <

oo includes k, £, M, and T. With this, we may then take covariant deriva-

tives of the equation (23.38) and use the estimates for the covariant deriva-

tives of HN and rJ ·(see Lemma 23.10) to obtain bounds for l8f\l~ (DxPN) I

which yield the existence of Gk,e satisfying (23.43). D

2. Existence of the heat kernel on a closed Riemannian manifold

via parametrix

Let (Mn, g) be a closed Riemannian manifold. Now that we have con-

structed a parametrix for its heat operator, we may use a space-time con-

volution to obtain a fundamental solution to the heat equation. We begin

with some preliminaries.

2.1. Space-time convolution and its general properties.

Given two functions

F,G E C^6 (M x M x (O,oo)),