The Spectrum of the Laplacian under Deformations of Homogeneous Metrics

Abstract: The spectrum of the Laplace-Beltrami operator is a key geometric feature of a compact Riemannian manifold, reflecting the subtle connections between analysis, geometry, and symmetry. While classical representation-theoretic methods yield explicit formulas for symmetric and normal homogeneous spaces via the Casimir operator, continuous metric deformations typically break these standard algebraic structures.

In this talk, we discuss a framework to compute the Laplace-Beltrami spectrum under multi-parameter deformations, focusing on canonical variations of homogeneous principal bundles. By algebraically realizing these deformed metrics as naturally reductive spaces on larger product groups, we reduce the analytical eigenvalue problem to a tractable algebraic branching problem. We apply this method to derive explicit formulas for the spectra of the classical families of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds and Stiefel manifolds. Finally, we illustrate how these explicit formulas can be used to study variational problems in global analysis, such as the linear stability of the associated Einstein metrics under Perelman's $\nu$-entropy and the occurrence of symmetry-breaking Yamabe bifurcations.

This is joint work with Ilka Agricola and Leandro Cagliero.

differential geometryrepresentation theoryspectral theory

Audience: researchers in the topic

Virtual seminar on geometry with symmetries

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

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