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Communications in Mathematical Physics

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

On Nodal Sets for Dirac and Laplace Operators

  • Christian Bär

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.

Keywords

Manifold Riemannian Manifold Dirac Equation Laplace Operator Differential Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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