Communications in Mathematical Physics

, Volume 188, Issue 3, pp 709–721 | Cite as

On Nodal Sets for Dirac and Laplace Operators

  • Christian Bär


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.


Manifold Riemannian Manifold Dirac Equation Laplace Operator Differential Form 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Christian Bär
    • 1
  1. 1.Mathematisches Institut, Universität Freiburg, Eckerstraße 1, D-79104 Freiburg, Germany.¶E-mail: baer@mathematik.uni-freiburg.deDE

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