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

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• 作者：Megan E. Gier ; Peter D. Hislop
• 关键词：Hodge Laplacian ; Eigenvalues ; Multiplicity ; Forms ; De Rham complex
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：26
• 期：4
• 页码：3176-3193
• 全文大小：475 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X
• 卷排序：26

文摘

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.