## Introduction Let $M$ be a compact Riemannian manifold with a metric $g$ and let $\lambda_1 \le \lambda_2\le\ldots$ be the eigenvalues of the [[The Laplace-Beltrami Operator|Laplace-Beltrami operator]] on it. Then we can define the *heat trace* $h_M(t) = \sum_{k=1}^\infty e^{-\lambda_k t}$which (as we will show below) converges for all $t\in(0, \infty)$. The reason the heat trace $h_M(t)$ is a particularly useful spectral invariant is because it determines the entire spectrum. If we write $\mu_1 < \mu_2 < \ldots$ to be the *distinct* eigenvalues with multiplicities $m_1, m_2, \ldots$, then $h_M(t) = \sum_{k=1}^\infty m_ke^{-\mu_kt}$ Now consider $\lim_{t\to\infty}e^\mu h_M(t)$. If $\mu < \mu_1$, then all the exponents in the sum are still negative and the limit goes to 0. If $\mu > \mu_1$, then the first exponential $m_1e^{(\mu_1-\mu)t}$ has a positive exponent and blows off to infinity. Only when $\mu = \mu_1$ does the first exponential remain constant, and $\lim_{t\to\infty} e^{\mu_1}h_M(t) = m_1$. Thus $\mu_1$ is the smallest value of $\mu$ that makes $\lim_{t\to\infty}e^\mu h_M(t)$ take on a finite nonzero value, while the multiplicity $m_1$ is that finite nonzero value. Once $\mu_1$ has been determined, we can determine $\mu_2$, and so on. Thus, the heat trace completely determines the spectrum. ## The heat kernel To understand where the heat trace comes from, its name, and why it converges, let us consider a different object for a moment. A *heat kernel* on $M$ is a function $e(t, x, y), x, y \in M, t \in \mathbb R_{> 0}$ that is a solution for the *heat equation* $\partial_tf(t, x) = \Delta_xf(t, x)$ with the initial heat distribution $f(0, x) = \delta_y(x)$ where $\delta$ denotes the Dirac-delta distribution. What this *means* is that the heat kernel describes the distribution of heat at time $t$ and position $x$ if initially "all" the heat was placed at position $y$[^1]. The heat kernel is also called the *fundamental solution* of the heat equation because if we want to solve the heat equation with some arbitrary initial heat distribution $f(0, x) = \varphi(x)$, we can use the fundamental solution to get a general solution via $f(t, x) = \int_Me(t, x, y)\varphi(x) dy$which is to say that we can "average out" the fundamental solution along the initial distribution to get an arbitrary solution. The main analytical black box that makes everything work is the following result on a *compact* Riemannian manifold: 1. the heat kernel always exists 2. It is unique 3. The following series converges $e(t, x, y) = \sum_{j=0}^\infty e^{-\lambda_jt}u_j(x)u_j(y)$ where $\lambda_j$ is the $j$th Laplacian eigenvalue and $u_j$ the corresponding eigenfunction such that $\{u_j\}$ is an orthonormal basis of eigenfunctions. We now see where the name heat trace comes from: define the *heat trace* $h_M(t)$ to be $h_M(t) = \operatorname{Tr}(e^{\Delta_M})=\sum_{i=0}^{\infty}e^{\lambda_kt}$ then we can see[^2] $h_M(t) = \int_Me(t, x, x)dx$from which we also get the convergence of the heat trace. Intuitively we can think of $e(t, x, x)$ as measuring the amount of heat left at point $x$ at time $t$ if all the heat started at $x$, and integrating over the manifold gives an idea of how fast heat disperses over the manifold. ## Heat Invariants In 1949 Minakshisunadaram and Pleijel proved the following asymptotic expansion for $e(t, x, x)$ as $t\to 0^+$: $\lim_{t\to 0^+}e(t, x, x) = (4\pi t)^{-\frac d2}\left(\sum_{j=0}^k a_j(x)t^j + O(t^{k+1})\right)$ for any $k>0$. These $a_j(x)$ are called *local heat invariants* and depend on the local geometry of $M$ near $x$. Now, define $a_j := \int_M a_j(x)dx$ called the (global) *heat invariant*. Recalling that $h_M(t) = \int_M e(t, x, x)dx$ we get the heat trace asymptotic expansion: $ \lim_{t\to0^+}h_M(t) = (4\pi t)^{-\frac d2}\sum_{j=0}^\infty a_jt^j$ Since $h_M(t)$ depends only on the spectrum of a manifold, it is a spectral invariant. Moreover, [[Calculating the heat invariants|calculations]] show that $a_0(x) = 1$ so $a_0 = \operatorname{Vol}(G)$ and $a_1(x) = \frac16 K(x)$ where $K$ is the local scalar curvature, so by integrating $a_1 =$ the total scalar curvature of $M$. Among other things, we instantly get that we can hear the Euler characteristic of a Riemann surface because (by Gauss-Bonnet): $\chi(M) = \frac1{2\pi}\int_MK(x)dx = \frac{3}\pi a_1$ ## References 1. I mostly follow the treatment of the heat trace done in *Topics in Spectral Geometry* by Levitin, Mangoubi and Polterovich. A preliminary version of the book is available for free on Levitin's website at https://michaellevitin.net/Book/ 2. Yaiza Canzani's notes on [Analysis on Manifolds via the Laplacian](https://www.math.mcgill.ca/toth/spectral%20geometry.pdf) give a more thorough treatment of the analytical side of constructing the heat kernel, showing that the heat trace determines the eigenvalues, etc. Note that in these notes, the heat trace is called the "Zeta function" as discussed in $\S7.8$ [^1]: Since we're not actually defining the heat kernel at $t=0$ this initial distribution can be phrased in a limit sense, i.e., $\lim_{t\to0^+}\int_M e(t, x, y) f(x) dx = f(y)$ which is precisely how the Dirac-delta distribution is defined. [^2]: Since $u_j(x)$ is an orthonormal basis, $\int_M u_j(x)^2dx = 1$