Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–klein 3-Folds

This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian 3-manifolds, namely nontrivial principal S¹ bundles P → X over Riemann surfaces equipped with certain S¹ invariant metrics, the Kaluza–Klein metrics. We prove for generic Kaluza–Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the S¹ action. We also construct an explicit orthonormal eigenbasis on the flat 3-torus T³ for which every non-constant eigenfunction has two nodal domains.