2 Heat Kernel for the Laplacian
In the previous section, our solutions had the property that in . One might hope that, for example, if is continuous, then this converges in . Another shortcoming of our previous approach is that we have no control over what looks like as a function as the whole construction is manifestly global.
In this section, we will construct the heat kernel via an alternative method. This method involves starting with an initial “guess” of what the heat kernel should be. We then show that if our initial guess is “good enough”, then there is an iterative procedure that gives us the actual heat kernel. Moreover, this iterative procedure is sufficiently explicit that we can relate the asymptotic behaviour of as back to that of our original guess.
We will now restrict to the case of the Laplace–Beltrami operator, since we can use the explicit solution for as an inspiration for our initial guess. We fix a and consider the Laplace–Beltrami operator on .
For any , there exists a smooth section such that
for all continuous .
There is a constant and an exponent such that
for all .
For any other continuous section , the convolution
is a section of .
To motivate why one should care about the convolution, notice that Duhamel's principle essentially says that the heat kernel
for all . The condition that the convolution is is a technical point that will be useful later on.
for the geodesic distance between
. The very first naive guess for
might just be
This definitely satisfies (i). To see how good an attempt we made, we compute of this.
To do so, we fix a and express in geodesic polar coordinates about . Then there is a formula
where is the determinant of the metric (note that is defined at , but is not, hence the funny notation). So we find that
If we were in flat space, then , and so this is in fact a legitimate fundamental solution, as one would expect. However, when , then we can't quite bound this as .
To solve this, we introduce some correction terms, which are small enough to preserve the property (i). We shall find with such that
satisfies the equality
The precise shape of the right-hand side does not matter. It is just what happens to be convenient. The point is that the exponent of is , which is eventually positive. Then we simply have to take for large enough (note that this is different from the exponent in (ii) by ). Actually, we will only succeed in finding in a neighbourhood of the diagonal, since we work in geodesic polars. But we can simply take , where is a bump function supported near the diagonal. ((iii) is left as an exercise)
To find the , we assume is of the given form, and compute
To obtain the desired result, we need the coefficient of to vanish for . So we need to inductively solve
The diligent reader will observe that the left-hand side is exact, so we can write this as
The existence and uniqueness of a local solution then follows easily (one has to be slightly careful about , where the original equation gives an initial condition ).
For future reference, we note that
where is a rational function in and its derivatives.
We will now fix an large enough and take to be the function given by the previous lemma.
To find the actual fundamental solution, we have to understand the convolution a bit better. Recall that we expect a genuine fundamental solution to satisfy . Using the fact that as , we can calculate
So at least formally, the heat kernel should be given by
Barring convergence issues, the formula above also shows that .
To check convergence, the naive bound
is not quite enough. We set
We then have . We claim that
The terms in the denominator are the interesting bits that come from doing the bound more carefully. They ensure the sum above converges by the ratio test. This is a standard inductive calculation
Adding these up and using the ratio test, we find that
converges, and we have the following theorem:
There is a fundamental solution , and it satisfies
In particular, we have
From this, we can read off the leading term of as , which is, unsurprisingly, . As a recap, to produce this estimate, we used this as an initial approximation to the heat kernel, and then perform some procedures to turn this into an actual heat kernel. We then observe that all the correction terms we add are of higher order, so we are happy.
Note that all we used about was the properties stated in Lemma 2.1, and the procedure is pretty general. One can interpret the success of this procedure as testifying to the locality of the heat kernel. The properties of we required were always local conditions, namely that looks locally like a heat kernel for small . We then get to produce an actual heat kernel , whose difference from is something we can control very well, and in particular vanishes in the limit .
In general, if our manifold is written as a union of two open submanifolds, and we have produced heat kernels on each submanifold (suitably interpreted), then we can attempt to glue them together using a partition of unity. This will not give us an actual heat kernel, but the failure to be a heat kernel decreases exponentially as (since falls exponentially as whenever ). The above argument then lets us produce a heat kernel whose difference from the “fake” one vanishes to all orders as .