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

Junya Takahashi

doi:10.1023/A:1021294732338

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

Junya Takahashi

INF - System Information Sciences

Research output: Contribution to journal › Article › peer-review

5 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 g_i (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 language	English
Pages (from-to)	13-27
Number of pages	15
Journal	Annals of Global Analysis and Geometry
Volume	23
Issue number	1
DOIs	https://doi.org/10.1023/A:1021294732338
Publication status	Published - 2003 Mar 1

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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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.",

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