1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

224 23. HEAT KERNEL FOR STATIC METRICS

on Minj(g) x JR}. Moreover, if N > ~ + k + 2£, then

(23.33)

af\J~ (DxHN) (x, t; y, u)

• (4 )-n/2 (t - )N-(n/2)-k-2£ exp (-d2 (x, y)) F (x y t - u)
  _ 1r u 4 (t-u) k,£ .. , ' '

where fA,e is a C^00 covariant k-tensor on Minj(g) x [O, oo).

PROOF. Since M is closed, there exists a finite collection of local coor-
dinate charts { (Ua, { x~} ~= 1 )} :=l and compact subsets Ka C Ua such that
m

LJ Ka = M. In each chart (Ua, { x~}) we may rewrite the components

a=l

of the covariant derivatives af\J~ (DxHN) (^8 i 1 , ... ,^8 ik) in terms of the
8xa 8xa

partial derivatives afat (DxHN) for 1 ::; j ::; k and the Christoffel symbols

and their derivatives. Now:

(1) From (23.29) we may deduce (23.33).

(2) From (23.31) in each Ka we deduce (23.32). D

The next step is to multiply the good approximation by a cutoff function
to obtain the so-called parametrix.

1.4. Existence of a parametrix for the heat operator - multi-

plying the good approximation by a cutoff function.

We formally define what it means for a space-time function to be a good
approximation to the heat kernel.

DEFINITION 23.11 (Parametrix for the heat operator). We say that a
C^00 function P : M x M x lRS,. -+ JR is a parametrix for the heat operator
~ - 2._ 8t if

(1) the functions (~x - gt) P and (~y + ffu) P both extend continu-

ously to M x M x JRS,. and

(2) limt\,uP(-,t;y,u) = Oy and limu)"tP(x,t; ·,u) =Ox, that is, for

any function f E C^0 (M),

(23.34a)

(23.34b)

lim { P(x,.t;y,u)f(x)dμ(x)=f(y),

t\,u}M

lim r p (x, t; y, u) f (y) dμ (y) = f (x).

u)"t}M.

Let (Mn, g) be a closed Riemannian manifold; note that since M is

closed, inj (g) ::; diam (g) < 09. Let HN: Minj(g) x JRS,. -+JR be as defined in

the previous section. Now we multiply the locally defined function HN by