The Multiplicity of Eigenvalues of the Hodge Laplacian on 5-Dimensional Compact Manifolds

Megan E. Gier; Peter D. Hislop

doi:10.1007/s12220-015-9666-7

The Multiplicity of Eigenvalues of the Hodge Laplacian on 5-Dimensional Compact Manifolds

Megan E. Gier
, Peter D. Hislop

Mathematics

Producción científica: Article › revisión exhaustiva

1 Cita (Scopus)

Resumen

We study the multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that the Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.

Idioma original	English
Páginas (desde-hasta)	3176-3193
Número de páginas	18
Publicación	Journal of Geometric Analysis
Volumen	26
N.º	4
DOI	https://doi.org/10.1007/s12220-015-9666-7
Estado	Published - oct 1 2016

Nota bibliográfica

Publisher Copyright:
© 2015, Mathematica Josephina, Inc.

Financiación

Both authors were partially supported by NSF Grant DMS 11-03104 during the time this work was done. We thank the referees for useful comments. This paper is partly based on the dissertation submitted by the first author in partial fulfillment of the requirements for a PhD at the University of Kentucky.

Financiadores	Número del financiador
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS 11-03104

ASJC Scopus subject areas

Geometry and Topology

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abstract = "We study the multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that the Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.",

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AU - Hislop, Peter D.

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N2 - We study the multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that the Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.

AB - We study the multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that the Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.

KW - De Rham complex

KW - Eigenvalues

KW - Forms

KW - Hodge Laplacian

KW - Multiplicity

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JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

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The Multiplicity of Eigenvalues of the Hodge Laplacian on 5-Dimensional Compact Manifolds

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ASJC Scopus subject areas

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