The exponent α=(n-2)/n is the critical threshold for uniform eigenvalue bounds of the weighted Laplacian with weight ρ^α: bounded below it, unbounded above it on manifolds of revolution.
Eigenvalues of the L aplacian on a compact manifold with density
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Eigenvalues of the Laplacian with density
The exponent α=(n-2)/n is the critical threshold for uniform eigenvalue bounds of the weighted Laplacian with weight ρ^α: bounded below it, unbounded above it on manifolds of revolution.