1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

342 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Lx,T· By definition, His the minimal positive solution to (26.6)-(26.7), i.e.,

(26.35a) ( ~~ - /::;.x, 7 H + Q (x, r) H) (x, r; y, v) = 0,

(26.35b) lim H ( ·, r; y, v) = 6y.

T',,V

We have the following symmetry property between H and the adjoint

heat kernel H* for L; r

'

LEMMA 26.13. For any x, y EM and 0 :S v < r :ST we have

H(x,r;y,v) = H* (y,v;x,r).

In particular,

(26.36a)

(26.36b)

( ~~ + /::;.y,vH) (x, r; y, v) + (R-Q) (y, v) H (x, r; y, v) = 0,

lim H (x, r; ·, v) = 6x.

V/'T

PROOF. This is true because of the construction of the heat kernel on

a noncompact manifold as the limit of a sequence of Dirichlet heat kernels

(see §5 of Chapter 24) and the fact that Lemma 26.3 holds for Dirichlet heat

kernels. D

We have the following LI estimate for the heat kernel in the noncompact

case.

LEMMA 26.14 (LI-norm of heat kernel is bounded-noncompact case).

Let (Mn,g (T)), r E [O, T], be an evolving complete noncompact Riemannian

manifold such that supM !sect (g (0))1, supMx[O,T] IRijl, supMx[O,TJ l'ViRjkl

are finite. Then for the minimal positive fundamental solution H to (26.4)

we have

(26.37)

for any y EM and 0 :S v < r :ST, where

CI ~ sup IR - QI < oo.

Mx[O,T]

Since CI = 0 when Q = R, we immediately obtain the following.

COROLLARY 26.15 (LI-norm of heat kernel is preserved when Q = R -

noncompact case). Under the hypotheses of Lemma 26.14, now with Q = R,

we have

(26.38)

for any y EM and 0 :S v < r :ST.