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

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 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

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

Access to Document

10.1023/A:1021294732338

Cite this

@article{fb3037d11aeb467a8c37ba095abb012e,

title = "On the Gap between the First Eigenvalues of the Laplacian on Functions and p-Forms",

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.",

keywords = "Collapsing of Riemannian manifolds, Comparison of eigenvalues, Laplacian on forms, Parallel forms, Spectrum",

author = "Junya Takahashi",

note = "Funding Information: ★ The author is partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.",

year = "2003",

month = mar,

doi = "10.1023/A:1021294732338",

language = "English",

volume = "23",

pages = "13--27",

journal = "Annals of Global Analysis and Geometry",

issn = "0232-704X",

publisher = "Springer Netherlands",

number = "1",

}

TY - JOUR

T1 - On the Gap between the First Eigenvalues of the Laplacian on Functions and p-Forms

AU - Takahashi, Junya

N1 - Funding Information: ★ The author is partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

PY - 2003/3

Y1 - 2003/3

N2 - 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.

AB - 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.

KW - Collapsing of Riemannian manifolds

KW - Comparison of eigenvalues

KW - Laplacian on forms

KW - Parallel forms

KW - Spectrum

UR - http://www.scopus.com/inward/record.url?scp=0037212186&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037212186&partnerID=8YFLogxK

U2 - 10.1023/A:1021294732338

DO - 10.1023/A:1021294732338

M3 - Article

AN - SCOPUS:0037212186

VL - 23

SP - 13

EP - 27

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 1

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this