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<p>We define a flexible class of Riemannian metrics on the three-torus. Then, using Stern’s inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1976_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song.</p>

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Stability for a Class of Three-Tori with Small Negative Scalar Curvature

  • 期刊论文

  • 发表日期：

    2025年4月2日

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  • Edward Bryden,
  • Lizhi Chen

摘要

We define a flexible class of Riemannian metrics on the three-torus. Then, using Stern’s inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in \(L^2\) L 2 norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song.