2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 231
THEOREM 23.16. For N > ~ and GN as in (23.55), the function
(23.56) H ~ PN + PN * GN
is contained in C^00 (M x M x (0, oo)), is independent of N, and satisfies
(23.57) DxH = DxPN + (DxPN) * GN - GN = 0.
Moreover, H is the unique fundamental solution to the heat equation.PROOF. STEP 1. H solves the heat equation. Let N > ~. By Corollary
23.21 below, the series G N = '2:::~ 1 (DxPN )*k in (23.55) converges abso-
lutely, where the convergence is uniform on compact sets. In particular,GN E C^0 (M x M x [O, oo)).
Since DxPN E C^0 (M x M x [O, oo)), this implies (see Exercise 23.22 below)
00 00
(DxPN) * L (DxPN)*k = L (DxPN)*k.
k=l k=2
Hence00 00
= DxPN + (DxPN) * L (DxPN)*k - L (DxPN)*k
k=l k=l
=0,
which is (23.57).STEP 2. H is C^00 • Furthermore, given £, m E NU {O} with 2£ + m <
N - ~'by Lemma 23.23 below we have that 8f8;1'GN exists, is continuous,
and
00
(23.58) afar;cN = :Lafar; ((DxPN)*k),
k=lwhere the convergence of the series on the RHS of (23.58) is absolute and
uniform on compact sets. In particular, GN ism+£ times differentiable in
the time and first space variables on M x M x [O, oo) for 2£ + m < N - ~.
In the next step we shall show that it follows, for any N > ~, that the
space-time function
00
(23.59) H = PN + PN GN = PN + PN L (DxPN)*k
k=l
is a fundamental solution to the heat equation.
Since the fundamental solution to the heat equation is unique (see The-
orem E.9 in Part II), we have that the expression on the RHS of (23.59)
is independent of N for N > ~. Since we may take N arbitrarily large,
from this independence of N, from PN E C^00 (M x M x ( 0, oo)), and from
PN*GN being m+f times differentiable in space-time when 2£+m < N - ~
(exercise), we conclude that His C^00 •