The Heat Kernel — The General Heat Equation

1.2 The General Heat Equation

We can further generalize our previous problem and replace Δ\Delta 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 MM and E,FE, F Hermitian vector bundles on MM. We are also given an elliptic differential operator D:Γ(E)Γ(F)D\colon \Gamma (E) \to \Gamma (F) over MM with formal adjoint D:Γ(F)Γ(E)D^*\colon \Gamma (F) \to \Gamma (E) (in the case of the Laplace–Beltrami operator, we have D=D=d+dD = D^* = \mathrm{d}+ \mathrm{d}^* and E=F=ΩE = F = \Omega ^*).

Similar to the case of the Laplace–Beltrami operator, we define

ΔE=DD:Γ(E)Γ(E)ΔF=DD:Γ(F)Γ(F). \begin{aligned} \Delta _ E & = D^* D \colon \Gamma (E) \to \Gamma (E)\\ \Delta _ F & = D D^*\colon \Gamma (F) \to \Gamma (F). \end{aligned}

Then kerΔE=kerD\ker \Delta _ E = \ker D and kerΔF=kerD\ker \Delta _ F = \ker D^*. So we have

indexD=dimkerDdimcokerD=dimkerDdimkerD=dimkerΔEdimkerΔF. \begin{aligned} \operatorname{index}D & = \dim \ker D - \dim \operatorname{coker}D \\ & = \dim \ker D - \dim \ker D^*\\ & = \dim \ker \Delta _ E - \dim \ker \Delta _ F. \end{aligned}

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

(t+ΔE)f=0 \left(\frac{\partial }{\partial t} + \Delta _ E\right) f = 0

on E×(0,)M×(0,)E \times (0, \infty ) \to M \times (0, \infty ), with t(0,)t \in (0, \infty ).

Hodge theory gives us an easy way to solve this, at least formally. We can decompose

L2(E)=λΓλ(E), L^2(E) = \bigoplus _\lambda \Gamma _\lambda (E),

where Γλ(E)\Gamma _\lambda (E) is the λ\lambda -eigenspace of ΔE\Delta _ E. The spectral theorem tells us the eigenvalues are discrete and tend to infinity. Let {ψλ}\{ \psi _\lambda \} be an orthonormal eigenbasis. Then we can write a general solution as

f(x,t)=λcλeλtψλ(x) f(x, t) = \sum _\lambda c_\lambda e^{-\lambda t} \psi _\lambda (x)

for some constants cλc_\lambda .

If we want to solve this with initial condition f0f_0, i.e. we require f(,t)f0f(-, t) \to f_0 as t0t \to 0 in L2L^2, then we must pick 1

f(x,t)=λeλtψλ(x)ψλ,f0. f(x, t) = \sum _\lambda e^{-\lambda t} \psi _\lambda (x) \langle \psi _\lambda , f_0\rangle .

Thus, we can write the heat kernel as

Ht(x,y;ΔE)=λeλtψλ(x)ψλ(y)T. H_ t(x, y; \Delta _ E) = \sum _\lambda e^{\lambda t} \psi _\lambda (x) \psi _\lambda (y)^ T.

Then we have

f(x,t)=MHt(x,y;ΔE)f0(y)  dy. f(x, t) = \int _ M H_ t(x, y; \Delta _ E) f_0(y)\; \mathrm{d}y.

We remark that for each fixed tt, the heat kernel Ht(x,y)H_ t(x, y) is a section of the exterior product EEM×ME \boxtimes E^* \to M \times M.

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.

Lemma 1.1

If Δ\Delta is any self-adjoint elliptic differential operator, then there is a constant CC and an exponent ε\varepsilon such that for large Λ\Lambda , the number of eigenvalues of magnitude Λ\leq \Lambda is at most CΛεC \Lambda ^\varepsilon .

Proof
To simplify notation, we assume Δ\Delta acts on the trivial line bundle. By replacing Δ\Delta with its powers, we may assume that the order dd of Δ\Delta is large enough such that we can apply the Sobolev (and regularity) bound fC0CfdC(ΔfL2+fL2). \| f\| _{C^0} \leq C' \| f\| _ d \leq C (\| \Delta f\| _{L^2} + \| f\| _{L^2}). In particular, if ψ\psi is an eigenfunction of eigenvalue at most Λ\Lambda , then we can bound ψC0C(1+Λ)ψL2. \| \psi \| _{C^0} \leq C(1 + \Lambda ) \| \psi \| _{L^2}. Let {ψλ}\{ \psi _\lambda \} be an orthonormal eigenbasis. Then for any constants aλa_\lambda and fixed xMx \in M, we have λΛcλψλ(x)C(1+Λ)(λΛcλ2)1/2. \left|\sum _{|\lambda | \leq \Lambda } c_\lambda \psi _\lambda (x) \right| \leq C(1 + \Lambda ) \left(\sum _{\lambda \leq \Lambda } |c_\lambda |^2\right)^{1/2}. We now pick cλc_\lambda to be the numbers ψˉλ(x)\bar{\psi }_\lambda (x), recalling that we have fixed xx. Then λΛψˉλ(x)ψλ(x)C(1+Λ)(λΛψˉλ(x)ψλ(x))1/2. \sum _{|\lambda | \leq \Lambda } \bar{\psi }_\lambda (x) \psi _\lambda (x) \leq C(1 + \Lambda ) \left(\sum _{\lambda \leq \Lambda } \bar{\psi }_\lambda (x) \psi _\lambda (x)\right)^{1/2}. Equivalently, we have λΛψˉλ(x)ψλ(x)C2(1+Λ)2. \sum _{|\lambda | \leq \Lambda } \bar{\psi }_\lambda (x) \psi _\lambda (x) \leq C^2(1 + \Lambda )^2. Integrating over all of MM, the left-hand side is just the number of eigenvalues of magnitude Λ\leq \Lambda . So we are done.

Elliptic regularity lets us bound the higher Sobolev norms of HtH_ t as well, and so we know HtH_ t is in fact smooth by Sobolev embedding.

It is fruitful to consider the “time evolution” operator etΔEe^{-t \Delta _ E} that sends f0f_0 to f(,t)f(-, t) defined above. This acts on the λ\lambda eigenspace by multiplication by eλte^{-\lambda t}. Thus, the trace is given by

ht(ΔE)tretΔE=λeλtdimΓλ(E). h_ t(\Delta _ E) \equiv \operatorname{tr}e^{-t \Delta _ E} = \sum _\lambda e^{-\lambda t} \dim \Gamma _\lambda (E).

The key observation is that DD gives an isomorphism between the λ\lambda eigenspace of ΔE\Delta _ E and the λ\lambda eigenspace of ΔF\Delta _ F as long as λ>0\lambda > 0, and the 00 eigenspaces are exactly the kernels of ΔE\Delta _ E and ΔF\Delta _ F. So we have an expression

indexΔE=ht(ΔE)ht(ΔF) \operatorname{index}\Delta _ E = h_ t(\Delta _ E) - h_ t(\Delta _ F)

for any t>0t > 0.

This expression is a very global one, because ht(ΔE)h_ t(\Delta _ E) depends on the eigenfunctions of ΔE\Delta _ E. However, the fact that eλΔEe^{-\lambda \Delta _ E} is given by convolution with Ht(x,y;ΔE)H_ t(x, y; \Delta _ E) gives us an alternative expression for the trace, namely

ht(ΔE)=MHt(x,x;ΔE)  dx. h_ t(\Delta _ E) = \int _ M H_ t(x, x; \Delta _ E)\; \mathrm{d}x.

This is still non-local, but we will later find that the asymptotic behaviour of Ht(x,x;ΔE)H_ t(x, x; \Delta _ E) as t0t \to 0 is governed by local invariants. Since the index is independent of tt, we can take the limit t0t \to 0 and get a local expression for the index.

Example 1.2

Take the flat torus Td=Rd/ZdT^ d = \mathbb {R}^ d / \mathbb {Z}^ d, and pick D=d:Ω0(M)Ω1(M)D = \mathrm{d}\colon \Omega ^0(M) \to \Omega ^1(M), so that Δ\Delta is the classical Laplacian

Δ=2xi2. \Delta = -\sum \frac{\partial ^2}{\partial x_ i^2}.

The eigenvectors are given by e2πixξe^{2\pi ix\cdot \xi } for ξZd\xi \in \mathbb {Z}^ d with eigenvalue 4π2ξ24\pi ^2 |\xi |^2. So the heat kernel is

Ht(x,y)=ξZde4πtξ2e2πi(xy)ξ=i=1dk=e4πtk2+2πi(xiyi)k. H_ t(x, y) = \sum _{\xi \in \mathbb {Z}^ d} e^{-4\pi t |\xi |^2} e^{2\pi i(x - y)\cdot \xi } = \prod _{i = 1}^ d \sum _{k = -\infty }^\infty e^{-4 \pi t k^2 + 2 \pi i (x_ i - y_ i) k}.

Number theorists may wish to express this in terms of the Jacobi theta function:

Ht(x,y)=i=1dϑ(xiyi;4it). H_ t(x, y) = \prod _{i = 1}^ d \vartheta (x_ i - y_ i; 4it).

We can explicitly evaluate the trace to be

ht=TdHt(x,x)  dx=(k=e4πtk2)d h_ t = \int _{T^ d} H_ t(x, x) \; \mathrm{d}x = \left(\sum _{k = -\infty }^\infty e^{-4\pi t k^2}\right)^ d

In the limit t0t \to 0, we can replace the sum with the integral to get

ht(e4πtk2  dk)d=(4t)d/2. h_ t \sim \left(\int _{-\infty }^\infty e^{-4\pi t k^2}\; \mathrm{d}k\right)^ d = (4t)^{-d/2}.