On the gap between the first eigenvalues of the Laplacian on functions and 1-forms

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概要

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• We study the first positive eigenvalue λ<SUB>1</SUB><SUP>(p)</SUP> of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the, inequality λ<SUB>1</SUB><SUP>(1)</SUP>≤λ<SUB>1</SUB><SUP>(0)</SUP> holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ<SUB>1</SUB><SUP>(1)</SUP><λ<SUB>1</SUB><SUP>(0)</SUP> holds. We show that any oriented closed manifold M with the first Betti number b<SUB>1</SUB>(M)=0 whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.

• 社団法人 日本数学会の論文

著者

• TAKAHASHI Junya

  Graduate School, Tohoku University

• Takahashi Junya

  Graduate School Of Mathematical Sciences

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• On the gap between the first eigenvalues of the Laplacian on functions and 1-forms
• Upper Bounds for the Eigenvalues of the Laplacian on Forms on Certain Riemannian Manifolds
• The First Eigenvalue of the Laplacian on $p$-Forms and Metric Deformations

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