Geometric Structure and the Laplace Spectrum

Abstract: The Laplace spectrum of a compact Riemannian manifold is defined to be the set of positive eigenvalues of the associated Laplace operator. Inverse spectral geometry is the study of how this set of analytic data relates to the underlying geometry of the manifold.

A (compact) geometric structure is defined to be a compact Riemannian manifold equipped with a locally homogeneous metric. Geometric structures played an important role in the study of two and three-dimensional geometry and topology. In dimension two, the only geometric structures are those of constant curvature. Furthermore, Berger showed that they are determined up to local isometries by their Laplace spectra.

In this work, we study the following question: “To what extend are the three-dimensional geometric structures determined by their Laplace spectra?” Among other results, we provide strong evidence that the local geometry of a three-dimensional geometric structure is determined by its Laplace spectrum, which is in stark contrast with results in higher dimensions. This is a joint work with Ben Schmidt (Michigan State University) and Craig Sutton (Dartmouth College).

differential geometrygeometric topologyspectral theory

Audience: researchers in the topic

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Virtual seminar on geometry with symmetries

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

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