Weyl's law is a fundamental result in spectral geometry that concerns the asymptotic behavior of the eigenvalues of the Laplace operator on a compact Riemannian manifold. It provides a connection between the geometry of the manifold and the distribution of its eigenvalues.
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For the positive scalar Laplace-Beltrami operator on a closed -dimensional Riemannian manifold, the number of eigenvalues at most , counted with multiplicity, has the displayed leading asymptotic. The exponent determines the dimension and the coefficient determines the total volume of the Riemannian metric. There is an analogous result with fixed Dirichlet boundary conditions or Neumann boundary conditions on a compact smooth domain.