Convergence of nodal sets in the adiabatic limit

TL;DR

This paper analyzes how the zero sets of eigenfunctions of the Laplacian behave in fibered manifolds as the fiber size shrinks, showing convergence to a limit set and relating it to the base manifold's eigenfunctions.

Contribution

It establishes the convergence of nodal sets in the adiabatic limit for fiber bundles with boundary and provides detailed descriptions for manifolds fibred over the circle.

Findings

Nodal sets converge to pre-images of a base eigenfunction's nodal set.

Nodal sets intersect the boundary for small fiber sizes.

Connected components of nodal sets are isotopic to fibers in the circle case.

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 $\pi{:}\, M\to S^1$.