MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{vol}(M)\ast A^n + o(\mathrm{something}) $$ (Perhaps one could consider first the case of a Kähler manifold? The Laplacian is particularly simple there.)

share|cite|improve this question
FYI, the most covariant "Laplacian" when the metric is not constant is the Laplace-Beltrami operator |g|^{-1/2} \partial_i |g|^{1/2} g^{ij} \partial_j (see I think that this agrees with yours, which depends on your choice of coordinates (but is the one I actually use) up to a first-order differential operator, and so the asymptotics should agree on compact manifolds. Oh, also, I don't think you should have a wedge? g^{ij} is symmetric in i,j, whereas \partial_i \wedge \partial_j looks antisymmetric? – Theo Johnson-Freyd Oct 19 '09 at 6:33
Ah, yes, it looks like the one I wrote in many, but not all, metrics. And I've written a corresponding symplectic form. My bad. – Ilya Nikokoshev Oct 19 '09 at 17:34
up vote 7 down vote accepted

The most natural way is to study the short-time asymptotics of the heat or wave kernel on M. For example, you can use the heat kernel $p_t(x,y) = \sum_i e^{-\lambda_i t} f_i(x) \overline{f_i(y)}$ where $f_i$ are the eigenfunctions with eigenvalues $\lambda_i$. This is a fundamental solution to the heat equation.

When $t$ is small then you can construct a good approximation to $p_t$ near any particular $x$ by hand, using Fourier analysis in local co-ordinates. The end result is that that $p_t(x,x) \approx C t^{-n/2}$. Now integrate this estimate $dx$, noting that $\int_M p_t(x,x)dx$ basically counts eigenvalues with $\lambda_i \leq 1/t$.

share|cite|improve this answer
For this proof a good reference (which is somewhat more general) is Chapter 2 of Heat Kernels and Dirac Operators by Berline, Getzler and Verne. See thm 2.41, which is a consequence of the asympotics of the heat kernel, thm. 2.30. – Simon Pepin Lehalleur Aug 9 '10 at 23:01

Wener Müler - Weyl laws...

has a pretty good introduction.

Edit: I see now that only the second section addresses you specific question and not the intro:(

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.