1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 361

Hence

d~(x,y)

c1e -cz(r-v).

H (x, T; y, v) ~ 1/2 1/2.

Vol 9 B9 (x, VT -v) Vol 9 B9 (y, VT -v)

(26. 77)

PROOF. Again let u (x, T) ~ H (x, T; y, v).

(1) By the Li-Yau inequality (25.58) with c: = !, we have (essentially

we have switched x and y in the first variable of (26. 75))

H (x, T; y, v) u(x, T)

H(y, T!v;y,v) u(y, T!v)

~ e_ciiT 2-2nexp (-cod~ (x,y)).

T-V

Then applying the lower bound (26.74) along the diagonal, we obtain

ce -C11T 2 2-2n exp ( -c~ o~ dE(x,y))

H (x, T; y, v) ~ (;=;;

Vol9B9 (y, y T)

const exp (-co~)

>

Vol9 B9 (y, VT -v)

The estimate with x and y switched on the RHS is proved analogously.

(2) Finally, (26.76) implies

2d~(x,y) 1

H(x ry v)2 > c2e-c2Cr-vl---~--~---~---~

' ' ' -^1 Vol9B9(x, VT -v) Vol9B9(y, .JT=V)'

which is (26.77). D

2.3. Double integral upper bound of the heat kernel for a fixed

metric.

The proof of the following special case of a result of Davies [52] uses
techniques similar to that in Theorem 26.25.

THEOREM 26.32 (Davies' double integral upper bound for the heat ker-

nel). If (Mn, g) is a complete Riemannian manifold and if H (x, y, t) is its

heat kernel, then for any two bounded subsets U1 and U2 of M we have

11

1 1 d^2 (U1,U2)

(26.78) H (x, y, t) dμ (x) dμ (y) ::; Vol2 (U1) Vol2 (U2) e- 4t ,

U1 U2.

where d (U1, U2) = infxEUi, yEUz d (x, y) is the distance between U1 and U2.

PROOF. Recall from Lemma 26.13 that H (x, y, t) = H (y, x, t). Let

Ui (x, t) ~ li H (x, y, t) dμ (y)