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 $$\#\{v < A^2\} = \mathrm{const}*\mathrm{vol}(M)*A^n + o(\mathrm{something})$$ (Perhaps one could consider first a case of Kahler manifold? The Laplacian is partiularly simple there).
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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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