Chapter 24. Heat Kernel for Evolving Metrics
I guess I'll never learn... another page is turned.- From "I'll Wait" by Van Halen
In this chapter we discuss the existence and asymptotics of the minimal
fundamental solution to the heat equation with a potential function and a
complete time-dependent metric (the manifold may be either compact or
noncom pact). Recall that the existence of the fundamental solution to the
adjoint heat equation with respect to a solution of the Ricci fl.ow, as well as
Perelman's Harnack-type inequality, was used in the proof of pseudolocality
in Chapters 21 and 22.
In §1 we discuss the existence of a parametrix for linear heat-type equa-
tions associated to a 1-parameter family of Riemannian metrics on a closed
manifold.
Similarly to the fixed metric case, in §2 we use this parametrix to prove
the existence of the heat kernel associated to a 1-parameter family of Rie-
mannian metrics on a closed manifold (we do not discuss all of the details
of the proof, which requires relatively minor modifications).
In §3 we modify the techniques in the fixed metric case to obtain an
asymptotic expansion for the heat kernel associated to an evolving metric
(for example, a solution to the Ricci flow).
In §4 we discuss aspects of the heat kernel asymptotic expansion related
to §9.6 of Perelman [152].
In §5 we consider the existence of heat kernels on noncompact manifolds
with evolving metrics as the limit of Dirichlet heat kernels for an exhaustion
by an increasing sequence of compact domains.
The setup in this chapter is the following. Let g (T), T E [O, T], be a
smooth 1-parameter family of complete metrics on a C^00 manifold Mn. We
write its evolution as
(24.1)where Rij is a general time-dependent symmetric 2-tensor (the notation Rij
is motivated by, but not to be confused with, the special case of the Ricci
tensor ~j ). Consider the heat operator with a potential term:
(24.2)[)L ~ 8T - b.T + Q'
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