- CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 217
(:IRS, c JR^2 is given by (23.1)) defined by
(^23. 7 ) E ( t ) , ( 4 (t _ ))-n/2 (-d
2
x, y, , u ...,... 7r u exp (x, y))
4 (t _ u).
Often we shall switch notation for the variables and writeE (x, t; y, u) = E (x, y, t, u).
In the case where (Mn, g) = lEn is Euclidean space, the function E is equal
to the Euclidean heat kernel.
Let inj (g) = infxEM inj 9 (x) E (0, oo) denote the injectivity radius of g
and let
(23.8) Minj(g) ~ {(x,y) EM x M: d(x,y) < inj (g)}.
Note that E restricted to Minj(g) x lRS, is C^00 since d^2 (x, y) is C^00 in Minj(g)
(while, on all of M x M x lRS,, we only have that E is Lipschitz).
In general, we think of E as a 'transplanted heat kernel', which, for
( x, y) near the diagonal of M x M and for t-u small, is a first approximation
to the heat kernel of ( M, g) that we are seeking to construct. In the next
subsection we construct a good approximation.
1.2. Constructing a good approximation to the heat kernel -
multiplying E by a finite series.
To improve our approximation to the sought after heat kernel on (Mn, g),
we multiply E by a finite series in the time variable with coefficients which
are functions on Minj(g)· In particular, for a given NE N with N > n/2,
we shall define a function
of the form^1
N
(23.9) HN (x, y, t, u) ~ HN (x, t; y, u) ~ E (x, t; y, u) L </>k (x, y) (t - u)k,
where the functions
<!>k : Minj(g) -+ JR,
fork= 1, ... , N, are to be defined below.
k=OOn Minj(g) x lRS,, we shall show that the function HN is a good approx-
imation to the heat kernel in a sense which we make precise in subsections
1.4 and 3.1 below. Note that since N > n/2,
0 < (t - u)N E (x, t; y, u) = (47r)-n/^2 (t - u)N-(n/^2 ) exp (-:~t(~ ~)
::::; ( 4 7r)-n/2 (t _ u)N-(n/2)tends to zero as t - u -+ 0, uniformly in x, y E M.
(^1) Since the Laplacian is with respect to a fixed metric, in the following discussion one
may take u = 0 for convenience.