1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

356 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

and there exists a constant C4 < oo depending only on T and sup IRij I such

that the minimal positive fundamental solution H to (26.4) satisfies^3

(

d~(x,y) )

C3 exp - C4(T-v)

(26.65) H(x,T;y,v) < ( ~) ( ~)

• ""IT vO 11/2 g B-g X, T-V • H 11/2 B- T-V
  2 vO g g y,^2

for any x, y EM and 0:::; v < T:::; T, where g is as in (25.5).

PROOF. Let DE (0, oo) be as in Lemma 26.23. By the inequality

d~ (x, z) d~ (z, y) (dg (x, z) + dg (z, y))^2 d~ (x, y)

~--+ > >-=---

T - p p - V - ( T - p) + (p - V) - T - V

for x, y, z EM and v < p < T and by the semigroup property (26.41), we
have

H(x,T;y,v)

=JM H (x, T; z, p) H (z, p; y, v) dμp (z)

d~(x,y) r d~(x,z) d~(z,y)

:::; e -2Jcr-v) j M H (x, T; z, p) e^2 D(r-p) · H (z, p; y, v) e^2 D(p-v) dμg(p) (z)

1 ·

d~(x,y) ( r d~(x,z) ) 2

:::; e -ZD(r-v) JM H^2 (x, T; z, p) eD(r-p) dμg(p) (z)

x (L H^2 (z,p;y,v)e~;"~d/'q(p) (z))! ,

where the last inequality follows from Holder's inequality. Substituting the
exponentially weighted integral bounds (26.61) and (26.62) into this, we
obtain

C ex p (-2D(T-v) d~(x,y) )
H(x,T;y,v):::; 1 ; 2 1 ; 2 '
Vol 9 B9(x,y!T-p)Vol 9 B9(y,ylp-v)
where C depends on n, T, K, and sup IRijl· The theorem now follows from
taking p = T!v. D

The following qualitative improvement of the previous theorem is Corol-

lary 5.2 in [26].

COROLLARY 26.26. The heat kernel H for L satisfies

H(x,T;y,v) ~Ce-~llmin·{

1

,

1

}·

. Vol9B9 ( x, ~) Vol9B9 (v, ~)

(^3) In fact, C4 = 2D, where D is as in Lemma 26.23.