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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.)

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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 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 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
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up vote 6 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 $\lambda_i \leq 1/t$.

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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
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Wener Müler - Weyl laws... http://www.math.uni-bonn.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:(

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