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Spectral multiplicity and nodal sets for generic torus-invariant metrics

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 $Δ_g$-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 $Δ_g$-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.