On nodal sets for Dirac and Laplace operators

Details On nodal sets for Dirac and Laplace operators On nodal sets for Dirac and Laplace operators

1997-07-10

arxiv.org/abs/dg-ga/9707008v1

Commun.Math.Phys.188:709-721,1997

Physics

High Energy Physics

High Energy Physics - Theory

LaTeX, uses pstricks macro-package, 15 pages with 2 figures; to appear in Commun. Math. Phys

Scientific paper

10.1007/s002200050184

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.

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