Convergence of nodal sets in the adiabatic limit

Abstract

We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on fibre bundles $\pi{:}\, M\to B$ in the adiabatic limit. This limit consists in considering a family $G_\varepsilon$ 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 $\varepsilon \ll 1$. 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 $F_x$ is independent of the base point. We prove for $\mathrm{dim} B \leq 3$ that the nodal set of the Dirichlet-eigenfunction $\varphi$ converges to the pre-image under $\pi$ of the nodal set of a function $\psi$ on $B$ that is determined as the solution to an effective equation. In particular this implies that the nodal set meets the boundary for $\varepsilon$ 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 $B=S^1$ we obtain finer estimates and prove that every connected component of the nodal set of $\varphi$ is smoothly isotopic to the typical fibre of $\pi{:}\, M\to S^1$.