1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

Chapter 23. Heat Kernel for Static Metrics

Got a good reason for taking the easy way out now.

• From "Day Tripper" by The Beatles

In this chapter we discuss the heat kernel on a compact manifold with a

fixed metric. In the next chapter we consider heat-type kernels on compact

and noncompact manifolds with time-dependent metrics. The two main

issues which we shall discuss are that of existence and asymptotic expansions

for short time.

Let

(23.1) lRS, = { (t, u) E JR.^2 : t > u}.

DEFINITION 23.1 (Fundamental solution to heat equation). Let (Mn, g)

be a complete Riemannian manifold. We say a function

h : M x M x lRS, -t JR

is a fundamental solution (to the heat equation) if

(1) h is continuous, C^2 in the first two space variables, and C^1 in the

last two time variables,

(2) changing notation by h (x, t; y, u) ~ h (x, y, t, u),

(23.2) ( :t -Llx) h ( ·, ·; y, u) = O, (! + Lly) h (x, t; ·, ·) = 0,

(3)

(23.3)

(23.4)

(23.5)

t'\,u lim h ( ·, t; y, u) = 8y, lim h (x, t; ·, u) = 8x,

u/t

i.e., for any continuous function f on M with compact support we

have

lim r h(x,t;y,u)f(x)dμ(x)=f(y),

t\,u}M

lim r h(x,t;y,u)f(y)dμ(y)=f(x).
u,/'t}M
REMARK 23.2. In fact, for this definition, we do not need to assume
that (M,g) is complete; however, for the applications we are interested in,
(M,g) shall be complete.
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