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

Megan E. Gier, Peter D. Hislop

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)3176-3193
Number of pages18
JournalJournal of Geometric Analysis
Issue number4
StatePublished - Oct 1 2016

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2015, Mathematica Josephina, Inc.


  • De Rham complex
  • Eigenvalues
  • Forms
  • Hodge Laplacian
  • Multiplicity

ASJC Scopus subject areas

  • Geometry and Topology


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