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<p>We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant <i>via</i> the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9996_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\frac{n}{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </msup> </math></EquationSource> </InlineEquation>-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions 5 and 6.</p>

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Rigidity of Einstein manifolds with positive Yamabe invariant

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月21日

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  • L. Branca,
  • G. Catino,
  • D. Dameno,
  • P. Mastrolia

摘要

We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant via the \(L^{\frac{n}{2}}\) L n 2 -norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions 5 and 6.