COMPLEX GEOMETRY Course notes

5.4 Heat equation approach
Given an initial distribution of heat f(x) = F (x, 0) (t = 0) on a Riemannian manifold (X, g), then the heat
F (x, t) at time t is governed by (∂ t + ∆ X )F = 0.
Example 5.4.1. F is easily obtained for every t for S 1 , and in general for the torus as follows:
F (0, t) = ∑ a n (t)e inθ .
We have
∂ t + ∆ θ = 0 =⇒ 0 = ∑ (
a
′
n (t) + n 2 a n (t) ) e inθ
=⇒ a n (t) = a n e −n2t , where a n = a n (0),
=⇒ F (θ, t) = ∑ e −n2t a n e inθ .
n≥0
It follows F (θ, t) −→ a 0 = ∫ S 1 f(θ)dθ = Av S 1(f), where the integral is the initial distribution.
Example 5.4.2. Let X = R. Doing the same exercise as above but using Fourier transforms, we get
F (x, t) = √ 1 ∫
∫
e − (x−y)2
4t f(y)dy = e R (x, y, t)f(y)dy.
4πt
R
The function e R (x, y, t) = 1 √
4πt
e − (x−y)2
4t is called the heat kernel. Similarly, e R n(x, y, t) = 1 √
4πt
e − ||x−y||2
4t .
On S 1 , we have e(x, y, t) = ∑ n e−n2t e in(x−y) = ∑ n e−n2t e inx e iny , where e −n2t are the eigenvalues of ∆, and
e inx and e iny are the eigenfunctions. In general, the existence of e X (x, y, t) is difficult to obtain analytically
but trivial on physical grounds.
R
Remark 5.4.1. F (x, t) is smooth for every t > 0, i.e., immediate smoothing by heat flow.
In general, given a form α on (X, g), wish to solve
{
(∂t + ∆)A(t) = 0
(∗)
A(0) = α
where α(t) is a form on X parametrized by t. Uniqueness of A(t) follows from:
Lemma 5.4.1. ||A(t)|| is decreasing (non-strict) for a solution of (∗).
Proof: ∂ t ||A(t)|| 2 = 2 〈∂ t A, A〉 = −2 〈∆A, A〉 = −2 〈 ||dA|| 2 + ||d ∗ A|| 2〉 ≤ 0.
68