Global analysis, analysis on manifolds Heat kernels, zeta functions and determinants Theorem

Minakshisundaram–Pleijel asymptotic expansion

Builds on Heat kernel on a Riemannian manifold Riemannian manifold Scalar curvature Spectrum of the Laplace–Beltrami operator

Statement

Let $(M, g)$ be a closed Riemannian manifold of dimension $n$ , and let $K(t, x, y)$ be its heat kernel. Then there are smooth functions $a_0, a_1, a_2, \ldots$ on $M$ such that for every integer $N \ge 0$ $K(t, x, x) = (4\pi t)^{-n/2} \left( \sum_{k = 0}^{N} a_k(x) \, t^k + O(t^{N+1}) \right) \quad \text{as } t \to 0^+,$ uniformly in $x \in M$ . Each $a_k(x)$ is a universal polynomial in the curvature tensor and its covariant derivatives at $x$ . The first two are $a_0 = 1$ and $a_1 = \tfrac16 S$ , where $S$ is the scalar curvature. Integrating over $M$ , the heat trace satisfies $\sum_{j \ge 0} e^{-t\lambda_j} \sim (4\pi t)^{-n/2} \sum_{k \ge 0} t^k \int_M a_k \, dv,$ where $\lambda_0 \le \lambda_1 \le \cdots$ is the spectrum of the Laplace–Beltrami operator.

Remarks

Idea of the proof: build an approximate heat kernel near the diagonal as the Euclidean Gaussian in normal coordinates times a power series $\sum u_k(x, y) t^k$ . The coefficients $u_k$ solve transport equations along geodesics, and $a_k(x) = u_k(x, x)$ . A Duhamel argument shows that the true heat kernel differs from the truncated approximation by $O(t^{N + 1 - n/2})$ . Consequences: the spectrum determines the dimension $n$ , the volume $\int_M a_0 = \operatorname{vol}(M)$ and the total scalar curvature $\int_M S$ . The next integrated coefficient is $\int_M a_2 = \frac{1}{360} \int_M \left( 2\abs{\mathrm{Rm}}^2 - 2\abs{\mathrm{Ric}}^2 + 5 S^2 \right) dv$ . Example: on a flat torus $\R^n / \Gamma$ , $K(t, x, x) = (4\pi t)^{-n/2} \sum_{\gamma \in \Gamma} e^{-\abs{\gamma}^2 / 4t}$ , so $a_0 = 1$ and $a_k = 0$ for $k \ge 1$ , with an exponentially small error.

Going deeper

For a student

Minakshisundaram and Pleijel used the expansion to continue the spectral zeta function of the Laplacian meromorphically. It shows that the spectrum determines the volume and the total scalar curvature. This answers some spectral rigidity questions. For example, a closed surface with the spectrum of the round sphere is a round sphere. The same construction works for Laplace-type operators on vector bundles, and the coefficients there are the Seeley–DeWitt coefficients. Combined with the McKean–Singer formula it leads to the heat equation proof of the index theorem.

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Heat kernel on a Riemannian manifold concept
Riemannian manifold concept
Scalar curvature concept
Spectrum of the Laplace–Beltrami operator concept
Minakshisundaram–Pleijel asymptotic expansion statement

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Tag	`45P2SEGGR1`
URI	`m2://45P2SEGGR1`
Kind	statement, theorem
Writing	W0 stub
MSC	58J35, 58J50, 35K08
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About	Heat kernel on a Riemannian manifold
Uses	Heat kernel on a Riemannian manifold, Riemannian manifold, Scalar curvature, Spectrum of the Laplace–Beltrami operator

Minakshisundaram–Pleijel asymptotic expansion

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

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