1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

258 23. HEAT KERNEL FOR STATIC METRICS

Then

Let H be the heat kernel on a closed Riemannian manifold (Mn, g).

From (23.132)-(23.134) we may deduce the following.

(1) If [so(x,y,t)[::; Cod(x,y), where Co< oo, then

(23.135) \JM sc(x,y,t)H(x,y,t)dμ(x)\::; Ct^112

fort small, where C < oo is independent of y.

(2) If [~ (x, y, t) / ::; Cid (x, y )^3 , where C1 < oo, then

(23.136) \JM~ (x, y, t) H (x, y, t) dμ (x)\ ::; Ct^312

fort small, where C < oo is independent of y.

(3) If {xi} are geodesic normal coordinates centered at y and if (Aij)

is a symmetric n x n matrix, then

(23.137)

I

r A-·xixj tr (A) I

1 JM x (x) iJ

4

t H (x, y, t) dμ (x) - -

2

• ::; Ct

for t small, where x : M ---+ JR. is any cutoff function equal to 1 in a

neighborhood of y with support in B (y, inj (g)) and where C < oo

is independent of y.

With this in mind, on a Riemannian manifold, let H = (47rt)-n/^2 e-f be

as in Theorem 23.16 and let 77 = 77 (d (x, y)) be the cutoff function defined