On the nodal sets of toral eigenfunctions

TL;DR

This paper investigates the conditions under which hypersurfaces can be contained in the nodal sets of Laplacian eigenfunctions on flat tori, revealing geometric restrictions and employing arithmetic and complex-analytic techniques.

Contribution

It establishes that hypersurfaces with nonzero Gauss-Kronecker curvature cannot lie on nodal sets for arbitrarily large eigenvalues, extending previous results to higher dimensions and using novel methods.

Findings

Hypersurfaces with nonzero Gauss-Kronecker curvature are excluded from nodal sets at high eigenvalues.

In 2D, segments of closed geodesics can lie on nodal sets for large eigenvalues.

The study employs arithmetic properties of lattice points and complex-analytic methods to analyze eigenfunction restrictions.

Abstract

We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional flat torus. The question we address is: Can a fixed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues. Our main result is that this cannot happen for hypersurfaces with nonzero Gauss-Kronecker curvature. In dimension two, the result follows from a uniform lower bound for the L^2-norm of the restriction of eigenfunctions to the curve, proved in an earlier paper. In high dimensions we currently do not have this bound. Instead, we make use of the real-analytic nature of the flat torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain. In all of our results, we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere. We also present an independent proof for the two-dimensional case relying on the "abc-theorem" in function fields.