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<p>For each degree <i>p</i> and each natural number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_10005_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(k \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we construct a one-parameter family of Riemannian metrics on any oriented closed manifold with volume one and the sectional curvature bounded below such that the <i>k</i>-th positive eigenvalue of the Hodge-Laplacian acting on differential <i>p</i>-forms converges to zero. This result imposes a constraint on the sectional curvature for our previous result in [<CitationRef CitationID="CR1">1</CitationRef>].</p>

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Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below

  • 期刊论文

  • 发表日期：

    2025年6月23日

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  • Colette Anné,
  • Junya Takahashi

摘要

For each degree p and each natural number \(k \ge 1\) k ≥ 1 , we construct a one-parameter family of Riemannian metrics on any oriented closed manifold with volume one and the sectional curvature bounded below such that the k-th positive eigenvalue of the Hodge-Laplacian acting on differential p-forms converges to zero. This result imposes a constraint on the sectional curvature for our previous result in [1].