1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

254 23. HEAT KERNEL FOR STATIC METRICS

we have

(23.121)

.. 8 (al/2)

-lJ.

8xJ

= (bij + ~Ripqj (y) xPxq + 0 (r^3 ))

x (~Rjq (y) xq +

2

~ (V'jRpq + 2\i'pRjq) (y) xPxq + 0 (r (x)^3 ))

= ~Rir (y) xr +

2

1

4

(Y'iRpq + 2\i'p~q) (y) xPxq + 0 (r (x)^3 ).

Thus, by (23.119) and by a-^1 a~i applied to (23.121) with a~i 0 (r (x)^3 ) =

0 (r (x)^2 ), we have

.6.x (a-1/2) = -a-1 _!!_, (gij 8 ( al~2))

8xi 8xJ

-1 -1

= a

6

~i + ~

2

(23.122) (2\i'iRiq + V' qRii) xq + 0 (r (x)^2 ),

where the Ricci tensors and their derivatives on the RHS are evaluated at y
and where we used

1 8

24

(\i'iRpq + 2\i'pRiq)

8

xi (xPxq)

1 1

=

24

(\i'iRiq + 2\i'iRiq) xq +

24

(Y'iilpi + 2\i' pRii) xP

1

=

12

(2\i'iRiq + V'qRii) xq.

LEMMA 23.32. The norm squared of the gradient and the Laplacian of

</>o = a-^1 /^2 are given by

(1)

(23.123)

(2)

(23.124)

I" ,.1-. 12 ( ) _ ij 8</>o 8</>o _^0 ( ( )

(^2) )
V Xl/-'0 X - g 8xi 8Xj - r X l
(.6.x</>o) (x) = .6.x ( a-^1!^2 ) (x) = R ~y) + ~ (Y'rR) (y) xr + 0 (r (x)^2 ).
PROOF. (1) By (23.120) we have
8</>o _ 1 q ( 2)
8 xj - f,Riq(y)x +0 r(x).

(2) This follows from applying to (23.122) the equations a= 1+0 (r^2 ),

gij = bij + 0 (r^2 ), and gij\i'iRjk = !VkR· D