Spectral multiplicity and nodal sets for generic torus-invariant metrics

TL;DR

This paper shows that for generic torus-invariant metrics on manifolds with free torus actions, eigenfunctions have simple symmetry properties and their nodal sets form connected hypersurfaces, revealing geometric and spectral structure.

Contribution

It establishes generic properties of eigenfunctions and eigenspaces under torus-invariant metrics, including irreducibility of eigenspaces and the topology of nodal sets.

Findings

Eigenpaces are irreducible real representations of the torus.

Nodal sets of certain eigenfunctions are connected hypersurfaces.

Eigenfunctions vanish on orbits have nodal sets dividing the manifold into two parts.

Abstract

Let a torus $T$ act freely on a closed manifold $M$ of dimension at least two. We demonstrate that, for a generic $T$-invariant Riemannian metric $g$ on $M$, each real $\Delta_g$-eigenspace is an irreducible real representation of $T$ and, therefore, has dimension at most two. We also show that, for the generic $T$-invariant metric on $M$, if $u$ is a non-invariant real-valued $\Delta_g$-eigenfunction that vanishes on some $T$-orbit, then the nodal set of $u$ is a connected smooth hypersurface whose complement has exactly two connected components.