1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. CONSTRUCTION OF THE PARAMETRJX FOR THE HEAT KERNEL 219

where 9ke ~ g (a/axk,a/axe). Therefore a is C^00 on Minj(g)· Moreover,

within the cut locus of y, the volume form may be expressed in positively

oriented normal coordinates as

dμ (x) =a (x, y) dx^1 /\ · · · /\ dxn.

REMARK 23.6 (Gauss lemma). Let x E M be a point within the cut

locus of y. If we let ei = 8 ~i (y), which is an orthonormal basis of TyM,

then we may write 8 ~i (x) = d (expY)exp;1(x) (ei)· The Gauss lemma says

that
a"(
\Jr (x) = ar (r (x))'

where 'Y: [O, r (x)] ---+ M is the unique minimal unit speed geodesic from y
to x. That is,

:' ~ !, (x) ~ ( °;; (r (x)), 0 !,) ~ (t, ~a!;' 0 !,) ~ t,: 9;;,

or equivalently,

(23.16)

n

xi= L,xj9ji·

j=l

Recall that the Laplacian, in the x variable, of a radial function

f(x,y)~J(r(x))

is given inside the cut locus Cut (y) (in particular, for points x E M such
that d (x, y) < inj (g)) by

_d^2 f a ~df

flxf - dr2 + ar log v <let g-dr

(23.17) = d

2

f + ( n - 1 + 8log a) df ,

dr^2 r ar dr

where Ir is the unit radial vector field, in the x variable, defined in Minj(g)

(note that Ir log J <let gs is the same as the mean curvature of the sphere
of radius r centered at y; see (1.132) and (1.135) in [45] for example).
Since

(

a^2 n-1 a a)

ar 2 + -r-ar - at E (. '. 'y' u) = O

for fixed (y, u) E M x (-oo, oo), inside the cut locus of y the heat operator

on M applied to E ( ·, ·, y, u) is given by

(

flx ~) E = 8loga aE = r 8loga E.

(^23 ·^18 ) at ar ar 2 (t - u) ar