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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds


Authors: Alberto Enciso and Daniel Peralta-Salas
Journal: Trans. Amer. Math. Soc. 364 (2012), 4207-4224
MSC (2010): Primary 58J50, 35B38
Published electronically: March 21, 2012
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Abstract: In this paper we analyze the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold $ M$. In particular, we show that the nonzero eigenvalues are simple and the zero set of the eigenforms of degree $ 1$ or $ 2$ consists of isolated points for a residual set of $ C^r$ metrics on $ M$, for any integer $ r\geq 2$. The proof of this result hinges on a detailed study of the Beltrami (or rotational) operator on co-exact $ 1$-forms.


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Additional Information

Alberto Enciso
Affiliation: Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
Email: aenciso@icmat.es

Daniel Peralta-Salas
Affiliation: Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain
Email: dperalta@icmat.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05496-1
PII: S 0002-9947(2012)05496-1
Received by editor(s): April 16, 2010
Received by editor(s) in revised form: September 24, 2010
Published electronically: March 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.