268 24. HEAT KERNEL FOR EVOLVING METRJCSAnalogous to (23.18), we have that in LJ 7 E(O,T] Minj(g(T)) x { r} x [O, r) C
M x M x JR~,- {) ( 2)
(
.6.x 7 _.!!_)E=8logaToE+ er r^7 E
' Or Or T Or T 4 ( r - V)(24.9)
(24.10)
r 7 u oga 7 rT7fT E
(£) 1 8r,,. )= -2(r-V) OrT +2(r-V) )
where a 7 ~ JdetgS (r) / r~-l and g^8 is as in (23.13). Note that a 7 (y, y) =
- REMARK 24.3. Although it may be difficult to calculate g 7 (r;) explic-
itly, we shall only need its qualitative properties. In particular, we have an
expansion of g 7 ( r; ( x)) for x near y.
Similarly to (23.10) and (23.11), we shall define the {'l/ik}f=o so that(24.11) 'l/io (y, y, r, v) = 1
and so that the approximate heat kernel HN satisfies the defining equation:
(24.12) Lx, N -7 (HN) (x, y, r, v) = (r - v) E (x, y, r, v) Lx, 7 ('i/iN) (x, y, r, v)
in LJ 7 E(O,T] Minj(g( 7 )) x { r} x [O, r), where
0
(24.13) Lx T ~ !:,) - .6.x T + Q.
' UT 'Given r E [O, T] and (x, y) E Minj(g( 7 ))' let VE TyM be the unit vector
tangent to the unique unit speed minimal geodesic from y to x with respect
to g ( r); call this geodesic
(24.14) '/1T,V: [O,rT (x)]-+ M.
The unit tangent (radial) vector to '/1T,V is 8 ~,,. ( s) = ( :f:s '}1 7 ,v) ( s).
Similarly to (23.23a)-(23.23b ), we define, recursively in k, the { 'l/ik}f=o
to solve the following first-order linear ODES along geodesics ')1 7 ,V emanating
from y with respect to the first variable in ( ·, y, r, v):
(24.15) O'l/Jo = (-~ OlogaT + ~ OrT) 'i/io
Or T 2 Or T 2 Or 'with the initial condition (24.11), and for 1 ~ k ~ N
( 24 16 ) rT O'l/Jk OrT + (rT 2 OlogaT OrT + k rT 2 OrT) Or nf, 'f'k _ - -L X,T (n!. 'f'k-l ) •
As in the fixed metric case, we have the following.LEMMA 24.4 (Recursive ODES for 'i/ik).
(1) With the initial data (24.11) and the assumption that the 'i/ik 's are
finite along the diagonal of M x M x JR~, the ODEs (24.15)-(24.16)