1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

334 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

We consider the heat(-type) equation

(26.4)

OU

Lx,TU ~ or - llg(T)U + Qu = 0,

where Q : M x [O, T] ---+ IR is a C^00 function and where b.. 9 ( 7 ) denotes the

Laplacian corresponding to g ( T), and we consider the corresponding adjoint

heat equation

(26.5) L~, 7 U ~ - ~~ - ll 9 ( 7 )U + (Q - R) u = 0.

Let H (x, r; y, v), where x, y E M and 0 :::; v < r s T, be the heat

kernel for L on a closed manifold, that is,

(26.6)

( ~~) (x, r; y, v) - (flx, 7 H) (x, r; y, v) + Q (x, r) H (x, r; y, v) = 0,

(26.7) limH(·,r;y,v)=Oy.

T'\,V

Equation (26.6) says that, with respect to the first pair of variables x and

r, H satisfies Lu= 0. Equation (26.7) says that for any¢ EC~ (M)

(26.8) lim r H(·,r;y,v)ef>(x)dμg(T)(x)=ef>(y).

T",,v}M

Define the adjoint heat kernel H* (x, r; y, v), where x, y E M and

0 :::; r < v :::; T, for L * by

(26.9)

(26.10)

( o! + llx, 7 ,H) (x, r; y, v) + (R-Q) (x, r) H* (x, r; y, v) = 0,

lim H* ( · ,r;y,v) = Oy.

T ,/'v

1.2. Elementary properties of the heat kernel and adjoint heat

kernel on a closed manifold.

The following is a space-time version of Green's second identity for the

operator Land its adjoint L*.^1

LEMMA 26.1 (Duhamel's principle on a closed manifold). Let g (r), r E

[O, T], be a time-dependent Riemannian metric on a closed manifold Mn. If

(^1) The space version of Green's second identity for the operator .6. says that if n c M
is a C^1 domain and if F, G E C^2 ( D), then
l (G.6.F -F.6.G) dμ =Ian ( G~~ - F~~) drY,
where v is the unit outward normal to aD; see (2.11) on p. 17 of Gilbarg and Trudinger
[71].