On Nodal Sets for Dirac and Laplace Operators

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. Mathematics Subject Classification: 58G03, 35B05 Keywords: generalized Dirac operator, Dirac equation, Laplace operator, nodal set, harmonic differential form 1 Introduction The motion of a vibrating membrane M fixed at the boundary is described by a function u : M \Theta R ! R satisfying the wave equation @ 2 u @t 2 + \Deltau = 0 and Dirichlet boundary ..