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

Enciso, Alberto; Peralta-Salas, Daniel

doi:10.1090/S0002-9947-2012-05496-1

Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds
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by Alberto Enciso and Daniel Peralta-Salas PDF

Trans. Amer. Math. Soc. 364 (2012), 4207-4224 Request permission

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 $r\geqslant 2$. 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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