1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

262 23. HEAT KERNEL FOR STATIC METRICS

PROOF. By the remark, for every s > 0 there exists Re:< oo independent

oft E [a,w] such that for every RE [Rc:,oo),

(23.143)

l

w r f ( x' t) dμ ( x) dt < c ( w - a).

a }M-B(O,R)

Let F (t) ~ JM f (x, t) dμ (x). Then (23.143) says for every s > 0 and
RE [Re:, oo),

1

w (F (t) - { f (x, t) dμ (x)) dt < s (w - a),

a }B(O,R)

so that

lim lw (F (t) - { f (x, t) dμ (x)) dt = 0.

R-+oo a j B(O,R)

That is,

l

w F(t)dt= lim lw { f(x,t)dμ(x)dt

a R-+oo a j B(O,R)

= lim r lw f (x, t) dtdμ (x)

R-+oo j B(O,R) a

=JM 1w f (x, t) dtdμ (x).

0

The following is Theorem 14 on p. 298 of Widder [187].

LEMMA 23 .40 (Differentiation under the integral sign). Let (Mn, g) be

a noncom pact Riemannian manifold. If

(i) f (x, t) E C^1 (M x [a, w]),

(ii) the improper integral JM f (x, t) dμ (x) converges uniformly int E

[a,w],

(iii) the improper integral JM~{ (x, t) dμ (x) converges uniformly int E

[a,w],

then

! JM f (x, t) dμ (x) =JM~{ (x, t) dμ (x).

PROOF. Let <p (t) ~JM~{ (x, t) dμ (x) and t E [a, w]. We compute

1£ <p (t) dt = 1£ JM~{ (x, t) dμ (x) dt

=JM 1t ~{ (x, t) dtdμ (x)

=JM f (x, £) dμ (x) - JM f (x, a) dμ (x).