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

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U2 - 10.1023/A:1021294732338

DO - 10.1023/A:1021294732338

M3 - Article

AN - SCOPUS:0037212186

SN - 0232-704X

VL - 23

SP - 13

EP - 27

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

IS - 1

ER -