1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

246 23. HEAT KERNEL FOR STATIC METRICS

and

\i'\i'd (ei, ej) = \i'\i'd (ei - (ei, V'd) \i'd, ej - (ej, V'd) \i'd)

1

= d (ei - (ei, V'd) \i'd, ej - (ej, V'd) V'd) + 0 (d)

1

= d (8ij - (ei, Y'd) (ej, V'd)) + 0 (d),

where we used \i'\i'd (X, Y) = ~ (X, Y) +0 (d) for unit vectors X and Y (see

(1.133) on p. 62 of [45] for example), which imply (23.100). This completes

the derivation of (23.99).

(I-2) Expanding 88? 8 Nj -^8 ?Nj. Now differentiating (23.81) twice with

x., Xx 8xz8Xz

respect to both the x and z variables and taking their difference while using

(23.98), we have (let u ~ t - s)^10

(23.101)

EPPN a^2 PN

axi x axj x 8xi z axj z

= _ 77 (d (x, z)) ( 4 7ru)-!j e_d

2

i::z) t (ad~ acp~ +ad~ acp~) (x, z) Uk

4u k=O ax~ ax1 ax1 ax~

+ 77(d(x,z)) ( 4 1fU )_!:!'. 2 e _d^2 4u (x,z);.... L...J (ad2 -. -. a¢k + -. ad^2 a¢k) -. ( X, Z ) U k

4u k=O ax1 ax{ ax{ ax1

• 77(d(x,z)) ( 4 7rU )_:!! 2 e _d^2 4u (x,z) (a^2 d^2 (x,z). - a^2 d^2 (x,z)). LN ,./.. 'f'k ( ) X, Z U k
  4u axi x ax^1 x axi z z ax^1 k=O

(d( ))( (^4) )_:!! _d
(^2) (x,z)...f!-..(a (^2) ¢k(x,z) a (^2) ¢k(x,z)) k

• 77 X, Z 1fU 2 e 4u L...J. -. U
  k=O ax~ax1 ax1ax{
  I _:!! _d^2 (x,z);.... (ad a¢k ad a¢k) k


• 77 (d (x, z)) (47ru) 2 e ---:r,;- 6 a i -j + -j ~ i (x, z) u
  k=O Xx axx axx uXx
  / (d( )) ( 4 )-:!! _d
  2

• 77 X, Z 1fU 2 e (x,z);.... 4u L...J (ad -. -. a¢k + -. ad -. a¢k) ( x, z U ) k
  k=O ax1 ax{ ax{ ax1

+ '(d( )) (a

2

d(x,z) a

2

d(x,z)) ( 4 )_:!! _d

2

77 X, Z. j -. j 1fU 2 e 4u (x,z) L...J ...f!-..,.J.. 'f'k ( x, Z ) U k.

ax~axx ax1axz k=O

(^10) Again the reader may choose to ignore the terms involving derivatives of 'T/ since
its support is away from the singularity. In (23.101) we have ordered the RHS so that the
terms involving 'T]^1 are at the end; by (23.98) no terms involving 'T/" appear.