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.)



Eigenvalues of Laplacian
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 $\lambda\_i \leq 1/t$. 


