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

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	3176-3193
Number of pages	18
Journal	Journal of Geometric Analysis
Volume	26
Issue number	4
DOIs	https://doi.org/10.1007/s12220-015-9666-7
State	Published - Oct 1 2016

Bibliographical note

Publisher Copyright:
© 2015, Mathematica Josephina, Inc.

Funding

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.

Funders	Funder number
National Science Foundation (NSF)	DMS 11-03104

Keywords

De Rham complex
Eigenvalues
Forms
Hodge Laplacian
Multiplicity

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s12220-015-9666-7

Cite this

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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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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.

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KW - Eigenvalues

KW - Forms

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KW - Multiplicity

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

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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