1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 221

(1) The defining equation (23.11) implies the following first-order lin-

ear ordinary differential equations along geodesics emanating from

y (corresponding to the coefficients in (23.22) being zero):

(23.23a)

o<f>o 1 Olag a

or (-'y) + 2 or </>o (. 'y) = o,

(23.23b) r ok or (·,y)+ (r8loga 2 or +k ) ¢>k(-,y)=b.xk-1(-,y).

for 1 ::::; k ::::; N.

(2) With the initial data (23.10) and the assumption that the ¢>k's are

finite along the diagonal of M x M, the ODEs (23.23a)-(23.23b)

on Minj(g) may be solved for smooth ¢>k recursively ink. Then the

function HN defined by (23.9) satisfies equation (23.11).

We now prove part (2) of the lemma. Equations (23.23a) and (23.10)
imply

(23.24) </>o (x, y) = a-^1 /^2 (x, y),

where we also used (23.14). Since a is C^00 (and nonzero), we have that

o E C^00 (Minj(g)) ·
Next we consider k 2: 1. We rewrite the ODE (23.23b) as

~ Or (rk al/2"' 'f'k ) = al/2rk-1 , b. X'f'k-1, "'

so that, for 1 ::::; k ::::; N, we have the recursive formula (using the requirement
that ¢>k is finite along the diagonal)

k 1/2 r(x) 1/2 k 1
(23.25) ¢>k (x, y) = r (x)- a- (x, y) Jo a r - b.xk-1dr,

where the integral is along the unique unit speed minimal geodesic joining
x toy and where d (x, y) < inj (g).

REMARK 23.8. Note that ¢k is independent of N 2: k.

Given (x, y) E Minj(g)' let VE sn-l c TyM be the unit vector tangent

to the unique minimal geodesic from y to x. Making the change of variables
p ~ rjr·(x), we may rewrite (23.25) as

LEMMA 23.9 (Recursive formula for the ¢>k). If for 1 ::::; k ::::; N the </>{; 's

are finite along the diagonal of M x M, then the solutions to (23.23b) are

given.by
(23.26)

¢>k (x, y) = a-^1 /^2 (x, y) fo

1

pk-l (a^1 l^2 b.xk-1) (expy (pr (x) V), y) dp

for (x,y) E Minj(g)·