Analysis and Mathematical Physics

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 $\mathbb{T}$ is a torus acting freely on $M$, the Laplace eigenspaces associated to a generic $\mathbb{T}$-invariant metric are irreducible representations of $\mathbb{T}$. 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 $\mathbb{T}$-invariant metric is an embedded hypersurface. In particular, under suitable conditions, our framework allows us to provide numerous examples of Riemannian manifolds with $S^{1}$-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 $L^{2}(M)$.

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