1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^216) 23. HEAT KERNEL FOR STATIC METRICS
We say that a fundamental solution H to the heat equation is the min-
imal positive fundamental solution if H is positive and if for every
positive fundamental solution h we have h 2: H. The minimal positive fun-
damental solution, which we shall show always exists, is also called the heat
kernel. By definition, the minimal positive fundamental solution is unique.
In this chapter we shall present a proof of the following classical result.

Riemannian manifold 1. Construction of the parametrix for the heat kernel on a

(Mn, g) is a closed Riemannian manifold, then there exists a fundamental

solution H (x, t; y, u) to the heat equation. Moreover, H (x, t; y, u) is unique,

positive, C^00 , symmetric in x and y, and

(23.6) JM H(x,t;y,u)dμ(x) = 1.

In particular, H (x, t; y, u) is the heat kernel.

REMARK 23.4. For this theorem we only discuss existence and smooth-

ness. Uniqueness, positivity, and symmetry were discussed earlier in Theo-

rem E.9 and Lemma E.11, both in Part II.

In the first three sections we discuss the existence of the heat kernel on

a closed Riemannian manifold.

In §1 we start with a good approximation to the heat kernel.

In §2 we construct the heat kernel by establishing the convergence of the

'convolution series'.

In §3 we prove some results on differentiating convolutions used in the

previous section.

In §4 we discuss aspects of the asymptotics of the heat kernel.

In §5 we recall some elementary facts used earlier.

via parametrix 2. Existence of the heat kernel on a closed Riemannian manifold

Riemannian manifold
In this section we discuss the construction of the 'parametrix' (good ap-
proximation) for the heat kernel on a closed oriented Riemannian manifold.
In the next section we use this to discuss the existence of the heat kernel.
The manner of our presentation is intended to facilitate the later adaptation
to the evolving metric case.
In this section (Mn, g) shall denote a closed Riemannian manifold and

d ( x, y) shall denote the Riemannian distance between points x and y in M.

1.1. First approximation to the heat kernel on a manifold -
transplanting the Euclidean heat kernel.

Consider the function

E: M x M x JR.S, -t (0, oo)