On the Gap between the First Eigenvalues of the Laplacian on Functions and p-Forms

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6 Citations (Scopus)

Abstract

We study the first positive eigenvalue λ 1(p) (g) of the Laplacian on p-forms for a connected oriented closed Riemannian manifold (M, g) of dimension m. We show that for 2 ≤ p ≤ m - 2 a connected oriented closed manifold M admits three metrics gi (i = 1, 2, 3) such that λ1(p) (g1) > λ 1(0) (g1), λ1(p) (g2) < λ1(0) (g2) and λ1(p) (g3) = λ1(0) (g3). Furthermore, if (M, g) admits a nontrivial parallel p-form, then λ1(p) ≤ λ1(0) always holds.

Original languageEnglish
Pages (from-to)13-27
Number of pages15
JournalAnnals of Global Analysis and Geometry
Volume23
Issue number1
DOIs
Publication statusPublished - 2003 Mar

Keywords

  • Collapsing of Riemannian manifolds
  • Comparison of eigenvalues
  • Laplacian on forms
  • Parallel forms
  • Spectrum

ASJC Scopus subject areas

  • Analysis
  • Political Science and International Relations
  • Geometry and Topology

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