1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^244) 23. HEAT KERNEL FOR STATIC METRICS
since t - s is bounded from above. However, from this estimate (contrast
with (23.88)) we cannot conclude that the integral
(23.95)
1
t1 {) EPPN.
8

. (x,z,t-s)G(z,y,s)dμ(z)ds
o M xi xJ
converges absolutely.
The idea for solving this issue is to integrate by parts, i.e., to use the
divergence theorem. To accomplish this, we first need to 'move' the par-
tial derivatives from the x variable onto the z variable. Recall that we
are assuming (U, {xi} ~= 1 ) are geodesic coordinates centered at x E U and
B (x, inj (g)) CU; in particular,

supp (PN (x, ·, s)) CU.

Let

[)2pN [)2 PN

.. (x, z, ·) = {).
8
[)xi {)xJ xi xJ. (x, z, ·)
x x
denote the original second partial derivatives with respect to the x variable

in the coordinates {xi} and let

82 PN.

-~-i-. {)--;-j ( x' z'. )

UXz Xz

denote the second partial derivative with respect to the z variable in the

same coordinates {xi} centered at x, which is well defined since z EU.

Since supp (PN (x, ·, s)) c U, we may write the expression in (23.94) as

1 (

{)^2 PN {)^2 PN )

.. - .. (x,z,t-s)G(z,y,s)dμ(z)

M [)xi x {)xJ x [)xi z 8xJ z

(23.96)

1

[)2pN

+.. (x, z, t - s) G (z, y, s) dμ (z).

M [)xi[)xJ z z

(I) We shall estimate the first term in (23.96) to show that the integral

1

t 1 ( [)2. PN j - [)2 .. PN ) (x,z,t-s)G(z,y,s)dμ(z)ds

.^0 M [)xi x 8x x [)xi z 8xJ z ·

(23.97)

converges absolutely.

(I-1) First and second derivatives of the distance function. Observe that

for (x, z) E Minj(g) such that x #-z

{) {)

(23.98) -

8

. d (x, z) + -
8
. d (x, z) = 0.
xi£ x~

Indeed, using d^2 (x, z) = ~J=l xJ (z)^2 , first we compute

{) 2.. {) (~. 2) i

{)xt d (x, z) = {)xt f:;;. xJ (z) = 2x (z),