1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

338 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Given y, z E M and 0 ::; v < p ::; T, using (26.6) and (26.15), we

compute for TE (v, p)

(26.22)

d~ (JM H (z, p; x, T) H (x, T; y, v) dμg(T) (x))

=JM '(~ OT (z,p;x,T)H(x,T;y,v)+H(z,p;x,T) ~ OT (x,T;y,v) ) dμ 9 ( 7 )(x)

+JM H (z, p; x, T) H (x, T; y, v) R (x, T) dμg(T) (x)

= -JM (l:lx,TH) (z,p;x,T)H(x,T;y,v)dμg(T) (x)

+JM H(z,p;x,T) (l:lx, 7 H) (x,T;y,v)dμg(T) (x)

= 0,

where we integrated by parts to obtain the last equality. Thus we have the

semigroup property of the heat kernel:

LEMMA 26.8 (Semigroup property on a closed manifold). Let H be the

heat kernel for L. Then for y, z EM and 0::; v < T < p::; T

(26.23) JM H (z, p; x, T) H (x, T; y, v) dμg(T) (x) = H (z, p; y, v).

PROOF. Using (26.22) and (26.7), we compute

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

= lim { H (z, p; x, O') H (x, O"; y, v) dμg(cr) (x)

cr\,,v} M

=H(z,p;y,v).

D

1.3. Elementary properties of the Dirichlet heat kernel on man-

ifolds with boundary.

Let g (T), T E [O, T], be a smooth 1-parameter family of Riemannian

metrics on a compact manifold Mn with nonempty boundary oM. Let

HD (x, T; y, v) denote the Dirichlet heat kernel for Lx,T- We have the follow-

ing symmetry property between HD and the adjoint Dirichlet heat kernel

H'D for L~, 7 •

LEMMA 26.9. For any x, y EM and 0::; v < T::; T we have

(26.24) HD (x,T;y,v) = Hb (y,v;x,T).