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On nodal sets for Dirac and Laplace operators

Christian Baer
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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 $Δ$-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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BibTeXRIS

Christian Baer. 1997-07-10. On nodal sets for Dirac and Laplace operators. https://doi.org/10.1007/s002200050184

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