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.)

$\begingroup$ FYI, the most covariant "Laplacian" when the metric is not constant is the LaplaceBeltrami operator g^{1/2} \partial_i g^{1/2} g^{ij} \partial_j (see en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator). I think that this agrees with yours, which depends on your choice of coordinates (but is the one I actually use) up to a firstorder 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? $\endgroup$ – Theo JohnsonFreyd Oct 19 '09 at 6:33

$\begingroup$ 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. $\endgroup$ – Ilya Nikokoshev Oct 19 '09 at 17:34
The most natural way is to study the shorttime 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 coordinates. 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 $\lambda_i \leq 1/t$.

$\begingroup$ 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. $\endgroup$ – Simon Pepin Lehalleur Aug 9 '10 at 23:01
Wener Müler  Weyl laws... http://www.math.unibonn.de/people/mueller/papers/weyllaw.pdf
has a pretty good introduction.
Edit: I see now that only the second section addresses you specific question and not the intro:(