It was unclear until recently whether the ergodicity of the geodesic flow on a given Riemannian manifold $M$ has any significant impact on the growth of the number of nodal domains of eigenfunctions of Laplace-Beltrami operator $\Delta_M$, as the eigenvalue $\lambda \to \infty$. In this talk, I'm going to explain my recent work with Steve Zelditch, where we prove that, when $M$ is a principle $S^1$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically $2$, independent of the eigenvalues. Note that principle $S^1$-bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when $(M,g)$ is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to $+\infty$ (proven in my paper with Seung Uk Jang).
I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. This provides a negative answer to the question raised by Thomas Hoffmann-Ostenhof:
``For any given orthonormal eigenbasis of the Laplace--Beltrami operator, can we always find a subsequence where the number of nodal domains tends to $+\infty$?''
