Spectral Multiplicity and Nodal Domains of Torus-Invariant Metrics

Abstract Let a $d$-dimensional torus $\mathbb{T}$ act freely and smoothly on a closed manifold $M$ of dimension $n>d$. We show that, for a generic $\mathbb{T}$-invariant Riemannian metric $g$ on $M$, each real $\Delta _{g}$-eigenspace is an irreducible real representation of $\mathbb{T}$ and, therefore, has dimension at most two. We also show that, for the generic $\mathbb{T}$-invariant metric $g$ on $M$, if $u$ is a non-invariant real-valued $\Delta _{g}$-eigenfunction that vanishes on some $\mathbb{T}$-orbit, then the nodal set of $u$ is a connected smooth hypersurface. If $n>d+1$, we show that the complement of the nodal set has exactly two connected components. As a consequence, we obtain new examples of manifolds for which—up to a sequence of Weyl density zero—each eigenfunction has exactly two nodal domains.