1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^252) 23. HEAT KERNEL FOR STATIC METRICS
of the metric has the expansion (see formula (23.139) below)
<let (gkc (x)) = 1-5kC}RkpqCXpxq - 5kC~\7rRkpq£Xpxqxr

• t (okC}RkpqCXpxq)
  2

• t (}RkpqCXpXq) (}RkrsCXrXs)

+ 0 kC( -^1 2 R )pqrs

20 \Jr \7sRkpqC + 45 Rkpqm Crsm X X X X

+o(r(x)^5 ),

so that (see Lemma 3.4 on p. 210 of [168])

<let (gkc( x))

= 1--R^1 xPxq - -\7^1 R xPxqxr

(23.114) 3Pq 6rpq

(

-^1 1 1 )pqrs
20 \7 r \7 sRpq + 90 RtpquRtrsu - lS RpqRrs X X X X

• 0 (r^5 ),
  where Rpq is the Ricci tensor.
  This formula implies

LEMMA 23.30.

a (x, y) = y/det (gkc) (x)

(23.115)

and so </>o has the expansion (which may be differentiated term by term)

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

(23.116) = 1+

1

1

2

Rpq (y) xPxq +

2

~ '7rRpq (y) xPxqxr + 0 (r (x)^4 ).

Next we consider the Laplacian of the distance (squared) function r^2 =

d^2 ( ·, y). By (23.17), the Laplacian of the distance function is given by^11

Therefore

(23.117)

(^11) Alternatively,
A _ n-l 8loga
D.xr---+ r a r.
Llx (r^2 ) (x) = ( 2 l\7rl^2 + 2rilxr) (x)

8loga

= 2n + 2r ( x) ar ( x)

(l:lxr) (x) = H (x) = ( :r log Jdetg~) (x),

where H is the mean curvature at x of the geodesic sphere S (p, r ( x)).