Generic Properties of Laplace Eigenfunctions in the Presence of Symmetry

Let G be a compact Lie group acting on a closed manifold M. Partially motivated by work of Uhlenbeck (1976), we explore the generic properties of Laplace eigenfunctions associated to G-invariant metrics on M. We find that, in the case where 𝕋 is a torus acting freely on M, the Laplace eigenspaces associated to a generic 𝕋-invariant metric are irreducible representations of 𝕋. This provides a mathematically rigorous instance of the belief in quantum mechanics that, in the presence of symmetry, non-irreducible eigenspaces are "accidental degeneracies.’’ We also observe that the nodal set of a Laplace eigenfunction associated to a generic 𝕋-invariant metric is an embedded hypersurface. In particular, under suitable conditions, our framework allows us to provide numerous examples of Riemannian manifolds with S1-invariant metrics for which the nodal set of any eigenfunction that is equivariant, yet non-invariant is connected and divides the manifold into precisely two nodal domains. Such eigenfunctions form a subspace of Weyl density one in L2(M).

This is joint work with Donato Cianci (GEICO), Chris Judge (Indiana) and Samuel Lin (Oklahoma).