$2$-nodal domain theorems for higher dimensional circle bundles

TL;DR

This paper extends 2-nodal domain theorems to higher-dimensional circle bundles, showing that certain eigenfunctions have connected nodal sets and exactly two nodal domains, generalizing previous 3D results.

Contribution

It generalizes earlier 3D results to higher dimensions for equivariant eigenfunctions on principal S^1 bundles, establishing connectedness and nodal domain counts.

Findings

Real parts of equivariant eigenfunctions have connected nodal sets.

Eigenfunctions have exactly 2 nodal domains.

Results fail for non-free S^1 actions, illustrated on spheres.

Abstract

We prove that the real parts of equivariant (but non-invariant) eigenfunctions of generic bundle metrics on nontrivial principal $S^1$ bundles over manifolds of any dimension have connected nodal sets and exactly 2 nodal domains. This generalizes earlier results of the authors in the $3$-dimensional case. The failure of the results on for non-free $S^1$ actions is illustrated on even dimensional spheres by one-parameter subgroups of rotations whose fixed point set consists of two antipodal points.