- BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 349
Given R > 0, let
NR (K) ~ { x EM : dg(O) (x, K) :'.SR}
denote the R-neighborhood of JC,. We shall assume that R is sufficiently
small so that NR (K) c int (0). Define PR: 0-+ [O, R] by
( ). { R-dg(O) (x, K) if x E NR (K),
PR X =
· o if x E o -NR (K).
Since IV PRl;(O) ::; 1 in 0 and g (r) ~ C0-^1 g (0) for r E [O, T], where Co ~
e^2 Tsupl'R-ijl < oo, we have
(26.53)
on 0 x [O,T].
Given 0, define
e: 0 x (O,min{T,<Y})-+ IR'.
by
'> c( x,r )=-. -PA(x)
2Co (<Y - r)We have that e is a Lipschitz function with
ae 1 2
OT + 21ve1g(T)PA (x) PA (x) IV PRl;( 7 )
- 2+ - 2
2Co (<Y - r) 2C;5 (<Y - r)
- 2+ - 2
(26.54) ::;^0
by (26.53).
Using (26.54), (26.51), and (26.50), we compute that
d~ Jn v^2 eedμg(T) = Jn ( 2v ~~ + v
2
~; + v2R) eedμg(T)
:S Jn (2v~v - ~v^2 1ve12- 2 ( Q +Co:__ ~R) v
2
) eedμg(T)(26.55) ::; -~ r l2VV + vve1 eedμg(T) + r 2V ~V eedμg(T)
2k Jen uv
::; 0since v = 0 on 80 x [O, T] (all we need is vg~ :S 0 on 80 x· [O, T]).
STEP 2. Upper estimate for IR (r). Let(26.56) IR (r) ~ r v^2 (x, r) dμg(T) (x).
ln-NR(K)