1.2 The General Heat Equation
We can further generalize our previous problem and replace by any elliptic self-adjoint operator. We will soon specialize to the case of the Laplace–Beltrami operator, but this section is completely general.
Our original motivation was to understand the index of operators, so let us start from there. Suppose we are given a compact Riemannian manifold and Hermitian vector bundles on . We are also given an elliptic differential operator over with formal adjoint (in the case of the Laplace–Beltrami operator, we have and ).
Similar to the case of the Laplace–Beltrami operator, we define
Then and . So we have
This is the form of the index that will be of interest to us.
By analogy with the classical heat equation, we consider the equation
on , with .
Hodge theory gives us an easy way to solve this, at least formally. We can decompose
where is the -eigenspace of . The spectral theorem tells us the eigenvalues are discrete and tend to infinity. Let be an orthonormal eigenbasis. Then we can write a general solution as
for some constants .
If we want to solve this with initial condition , i.e. we require as in , then we must pick
Thus, we can write the heat kernel as
Then we have
We remark that for each fixed , the heat kernel is a section of the exterior product .
To ensure the sum converges, we need to make sure the eigenvalues grow sufficiently quickly. This is effectively Weyl's law, but we for our purposes, we can use a neat trick to obtain a weaker bound easily.
If is any self-adjoint elliptic differential operator, then there is a constant and an exponent such that for large , the number of eigenvalues of magnitude is at most .
To simplify notation, we assume
acts on the trivial line bundle. By replacing
with its powers, we may assume that the order
is large enough such that we can apply the Sobolev (and regularity) bound
In particular, if
is an eigenfunction of eigenvalue at most
, then we can bound
be an orthonormal eigenbasis. Then for any constants
, we have
We now pick
to be the numbers
, recalling that we have fixed
Equivalently, we have
Integrating over all of
, the left-hand side is just the number of eigenvalues of magnitude
. So we are done.
Elliptic regularity lets us bound the higher Sobolev norms of as well, and so we know is in fact smooth by Sobolev embedding.
It is fruitful to consider the “time evolution” operator that sends to defined above. This acts on the eigenspace by multiplication by . Thus, the trace is given by
The key observation is that gives an isomorphism between the eigenspace of and the eigenspace of as long as , and the eigenspaces are exactly the kernels of and . So we have an expression
for any .
This expression is a very global one, because depends on the eigenfunctions of . However, the fact that is given by convolution with gives us an alternative expression for the trace, namely
This is still non-local, but we will later find that the asymptotic behaviour of as is governed by local invariants. Since the index is independent of , we can take the limit and get a local expression for the index.
Take the flat torus , and pick , so that is the classical Laplacian
The eigenvectors are given by for with eigenvalue . So the heat kernel is
Number theorists may wish to express this in terms of the Jacobi theta function:
We can explicitly evaluate the trace to be
In the limit , we can replace the sum with the integral to get