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Annals of Global Analysis and Geometry

, Volume 23, Issue 1, pp 13–27 | Cite as

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

  • Junya Takahashi
Article

Abstract

We study the first positive eigenvalue λ(p)1(g) of the Laplacian on p-forms for a connected oriented closed Riemannianmanifold (M, g) of dimension m. We show that for 2 ≤ pm − 2 a connected oriented closed manifold M admits three metrics g i (i = 1, 2, 3) such that λ(p)1(g1)> λ(0)1(g1),λ(p)1(g2) < λ(0)1(g2) andλ(p)1(g3)= λ(0)1(g3).

Furthermore, if (M, g) admits a nontrivial parallel p-form,then λ(p)1 ≤ λ(0)1 always holds.

Laplacian on forms spectrum comparison of eigenvalues collapsing of Riemannian manifolds parallel forms 

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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Junya Takahashi
    • 1
  1. 1.Mathematical InstituteTôhoku UniversityAoba, SendaiJapan

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