1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

Since

5. HEAT KERNEL ON NONCOMPACT MANIFOLDS

00

= L Mk+l (xo, r; z, cr)

k=l

= 2 L oo 1r1 av-aii (xo, r; w, p) Mk (w, p; z, cr) dμg(p) (w) dp

k=l er BM w,p

1

= 2 r1 -~-{}H (xo, r; w, p) M

00 (w, p; z, cr) dμg(p) (w) dp,

er BM uVw,p ,

we have that M 00 satisfies the equation

M 00 (xo, r; z, cr)

aii

= 2-~-(xo, r; z, cr)

uVz,cr

1

+ 2 r1 -oH.

0

• (xo, r; w, p) M 00 (w, p; z, cr) dμg(p) (w) dp.
  er BM Vw,p
  Finally, by (24.86) we note that

1

r 1 aii

00

U'lj; 00 (x, r) = - dcr a;;-(x, r; z, cr) L Ak (z, cr) dμg(cr) (z)

0 BM z,cr k=O

= - L oo 1r dcr^1 a;;-aii (x, r; z, cr) Ak-1 (z, cr) dμg(cr) (z)

k=l 0 BM z,cr

= - f r dcr { Mk (x,r;z,cr)b(z,cr)dμg(cr) (z)

k=l lo lBM

= - r dcr { M 00 (x, r; z, cr) b (z, cr) dμ 9 (cr) (z);

lo lBM

here we equated (24.82) and (24.88) to obtain the third equality.

5.3. Heat kernels on noncompact manifolds.

301

The existence of the heat kernel on noncompact manifolds is given by
the following; see Theorem 4 and its proof on pp. 188-191 of [27].

THEOREM 24.39 (Existence and uniqueness of heat kernel on noncom-
pact manifolds - fixed metric). Let (Mn, g) be a (not necessarily complete)
noncompact Riemannian manifold. There exists a unique minimal positive
fundamental solution HM (x, y, t) of the heat equation (also called the heat
kernel). Moreover, HM (y, x, t) is 000 and symmetric in x and y.

Now we consider the time-dependent metric case.