1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^218) 23. HEAT KERNEL FOR STATIC METRICS

Fix y E M. We shall show the existence of the functions { ¢k}f=o in

Lemma 23.7 with

(23.10) <Po (y,y) = 1

and so that HN satisfies the defining equation:

(

(23.11) /J.xHN - 8HN) at (x, t; y, u) = E (x, t; y, u) (!J.x<PN) (x, y) (t - u) N

in Minj(g) x JR.S,., where /J.x denotes the Laplacian with respect to the x

variable.

REMARK 23.5. The reason for the choice of the form of the RHS of (23.11)

will become apparent from the derivation of (23.23b) (note, in particular,

the case k = N of (23.23b)).

We now compute the ODE for the tPk which we derive from equation

(23.11) with the initial values (23.10).^2 In doing so, we shall see that tPk is

C^00 for 0 :::::; I~ :::::; N.

Since Eis a radial function in the space variable x centered at y, we con-

sider the equations for tPk (derived from the equation for HN) on geodesics

emanating from y. Given our y EM, let

r(x)~d(x,y)

and define the normal coordinates volume density (or Jacobian)

a : Minj(g) --+ (0, oo)

by

(23.12) a (x ) = .J det g

(^5) ( x)

,y. r (xr-1 '

where detg^5 (x) = det (gg) (x) and

(23.13) gg ~ g (a/ aei, a/ f)(Jj)

are the components of the metric in geodesic spherical coordinates {Bi} 7= 1

centered at y.^3 Note that

(23.14) lim a (x, y) = 1 ~a (y, y),

x--+y

so that a is well defined along the diagonal of Minj(g).

Let { xi}~~ 1 be exponential normal coordinates centered at y. We have

(23.15) a(x,y) = .jdet(gke)(x),

(^2) By the Ansatz (23.9), the expansion in powers oft-u for (23.11) termwise vanishes.
(^3) For a discussion of geodesic spherical coordinates, see subsection 10.2 of Chapter 1
in [45].