Restriction of toral eigenfunctions to hypersurfaces and nodal sets

TL;DR

This paper establishes uniform bounds for the restriction of Laplacian eigenfunctions on a 3D torus to curved surfaces and explores related properties of their nodal sets.

Contribution

It provides new uniform bounds for eigenfunction restrictions and insights into the structure of their nodal sets on the flat torus.

Findings

Uniform upper and lower bounds for eigenfunction restrictions

Results on the geometry of nodal sets

Implications for eigenfunction behavior on curved surfaces

Abstract

We give uniform upper and lower bounds for the L^2 norm of the restriction of eigenfunctions of the Laplacian on the three-dimensional standard flat torus to surfaces with non-vanishing curvature. We also present several related results concerning the nodal sets of eigenfunctions.