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.
Original language | English |
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Pages (from-to) | 13-27 |
Number of pages | 15 |
Journal | Annals of Global Analysis and Geometry |
Volume | 23 |
Issue number | 1 |
DOIs | |
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