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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Received: 28 October 1996 / Accepted: 3 March 1997
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Bär, C. On Nodal Sets for Dirac and Laplace Operators . Comm Math Phys 188, 709–721 (1997). https://doi.org/10.1007/s002200050184
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DOI: https://doi.org/10.1007/s002200050184