Using Hodge spectra to detect orbifold singularities

Abstract: A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold. A fundamental question in the Laplace spectral geometry of Riemannian orbifolds is whether or not a singular orbifold can be isospectral to a manifold. This question is open for the spectrum of the Laplacian acting on functions. We will see that combining information from the spectrum of the Laplacian on functions with information from the spectrum of the Hodge Laplacian on 1-forms allows us to detect orbifold singularities in some cases. For example, a singular Riemannian orbifold of dimension 3 or less cannot be both 0 and 1-isospectral to a Riemannian manifold. The proof relies on the heat invariants associated to the $p$-spectrum of the corresponding Hodge Laplacian. We will also discuss a few inverse spectral results for the individual $p$-spectra themselves.

differential geometryspectral theory

Audience: researchers in the topic

( paper | video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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