290 24. HEAT KERNEL FOR EVOLVING METRICS5. Heat kernel on noncompact manifolds
Thus far in this chapter and the previous chapter, we have discussed heat
kernels and parametrixes on closed manifolds. In this section we prove the
existence of heat kernels on noncompact manifolds via the limit of Dirichlet
heat kernels of smooth bounded domains.
5.1. Dirichlet heat kernels for fixed metrics.
First recall the following heat kernel existence result for the Dirichlet
problem; see Li [116] or Chapter VII of Chavel [27].THEOREM 24.30 (Existence and uniqueness of Dirichlet heat kernel). Let
(Mn, g) be a compact smooth Riemannian manifold with nonempty boundary8M. Then there exists a unique heat kernel HD (x, y, t) for the Dirichlet
problem (also called the Dirichlet heat kernel). That is, there exists a
unique continuous function HD : M x M x (0, oo) -+ [O, oo), which is C^00
on int (M) ><int (M) x (0, oo), such that
8
at HD (x, y, t) = b.xHD (x, y, t) in int (M) x int (M) x (0, CXl)'HD (x, y, t) = 0 on 8M x int (M) x (0, oo),
lim HD ( ·, y, t) = 8y for y E int (M),
t'\,0where int (M) denotes the interior of M.
More specifically, the Dirichlet heat kernel may be obtained as the con-
vergent series
00
HD (x, y, t) = L e->..kt'Pk (x) 'Pk (y),
k=l
where 'Pk is the k-th Dirichlet eigenfunction and Ak is its corresponding
eigenvalue, so that b.<pk +Aki.pk= 0, 'PklaM = 0, {<pk}~ 1 is an orthonormal
basis for L^2 (M, g), and0 < >.1 < >.2 :'.S A3 :'.S · • · •
See the beginning part of [116] for details.
5.2. Dirichlet heat kernels for evolving metrics.
Now we consider the time-dependent metric case. Let g ( T), T E [O, T], be
a smooth 1-parameter family of Riemannian metrics on a compact manifold
Mn with nonempty boundary 8M. We adopt the same setup as in the
beginning of this chapter: J 7 9ij ~ 2Rij and Lx, 7 ~ J 7 - b.x, 7 + Q, where
Q : M x [O, T] -+ IR is a C^00 function. ~
We extend M to a smooth compact manifold Mn without boundary and
we extend g (T), TE [O, T], to a smooth 1-parameter family of Riemannian
metrics g (T), T E [O, T], on M. For example, we may do this as follows.
First, extend the differentiable manifold M to a collar past its boundary.
Second, extend the metric so that in a collar of the new boundary it is