Convergence of nodal sets in the adiabatic limit

We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles π:M→B in the adiabatic limit. This limit consists in considering a family Gε of Riemannian metrics that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the base is given by ε≪1 . We assume M to be compact and allow for fibres F with boundary, under the condition that the ground state eigenvalue of the Dirichlet Laplacian on Fx is independent of the base point. We prove for dim(B)≤3 that the nodal set of the Dirichlet eigenfunction φ converges to the pre-image under π of the nodal set of a function ψ on B that is determined as the solution to an effective equation. In particular, this implies that the nodal set meets the boundary for ε small enough and shows that many known results on this question obtained for some types of domains, also hold on a large class of manifolds with boundary. For the special case of a closed manifold M fibred over the circle B=S1 , we obtain finer estimates and prove that every connected component of the nodal set of φ is smoothly isotopic to the typical fibre of π:M→S1