1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

348 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

LEMMA 26.21 (Exponential quadratically weighted L^2 -estimate in terms

of L^2 -estimate). Let (Mn' g ( T)) be as in (26.l). Let n c M be a smooth

compact domain and let /(, be a compact set with /(, C int ( n). There exists

C = C('Y,supJQJ,supJRJ) E (O,oo) and D = D('Y,T,suplRijl) with the

following property. Let u : n x [O, T] -+ JR be a solution to

OU

(26.47) OT = D..g(T)U - Q u

with Dirichlet boundary values

ulanx[O,T] = 0

and

supp ( u ( · , 0)) c K

in the sense that for x E n-K we have lim 7 \;o u (x, T) = 0 locally uniformly.

If

(26.48) Jn r u^2 (x, T) dμg(T) (x) ::::; f^1 (T)

for TE (0, T], where f is ("!, A)-regular, then

{ d~(o) (x,JC) 4AeCT

(26.49) Jnu

2

(x,T)e DT dμg( 7 )(x):::; f(T/"f)

for TE (0, T].

REMARK 26.22. An important special case is when /(, is a point and u

is a Dirichlet heat kernel centered at that point.

PROOF. STEP 1. Weighted L^2 estimate. Let Co > 0 be such that

(26.50) Co+ inf (Q -!R) 2: 0

nx[O,T] 2

and let

V ( X, T ) :::;:: · e -CoT U ( X, T ) •

Then (26.47) yields

(26.51)

ov

OT= D..v - (Q +Co) v

and (26.48) yields

1

e-2CoT 1

(26.52) v^2 (x,T)dμg(T)(x):::; f() ~---.

n T f (T)

Since f is ("!, A)-regular, we have that J (T) = e^2 Co^7 f (T) is also ("!,A)-
regular since J is strictly increasing and

! (T) = e2CoT(l--y-^1 ) f (T)

f(r/"f) f(T/"f)'

whereas e^2 CoT(l--Y-

1

) is nondecreasing in T.