On nodal sets for Dirac and Laplace operators

TL;DR

This paper investigates the geometric structure of nodal sets for solutions to Dirac and Laplace operators on Riemannian manifolds, establishing codimension bounds and regularity properties with optimal examples.

Contribution

It provides new proofs and results on the codimension and structure of nodal sets for Dirac solutions, Laplace eigenfunctions, and harmonic forms on Riemannian manifolds.

Findings

Nodal sets of Dirac solutions have codimension at least 2.

Nodal sets of Laplace eigenfunctions are smooth hypersurfaces with lower-dimensional singular sets.

Nodal sets of harmonic forms on closed manifolds have codimension at least 2.

Abstract

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension. We also see that the nodal set of a $\Delta$-harmonic differential form on a closed manifold has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal.