On the gap between the first eigenvalues of the laplacian on functions and 1-forms

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Abstract

We study the first positive eigenvalue λ(p)1 of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality λ(1)1≤λ(0)1 holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ(1)1(0)1 holds. We show that any oriented closed manifold M with the first Betti number b1(M)=0 whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.

Original languageEnglish
Pages (from-to)307-320
Number of pages14
JournalJournal of the Mathematical Society of Japan
Volume53
Issue number2
DOIs
Publication statusPublished - 2001

Keywords

  • Eigenvalue
  • Einstein manifold
  • Laplacian on forms
  • Stability

ASJC Scopus subject areas

  • Mathematics(all)

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