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<p>We prove (Theorem 1.1.) that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem obtained in [20] for the extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class. We also explore special cases of the quasi-Einstein equations related to integrability and the Hitchin equations, as well as to Einstein-Weyl structures and Kazdan-Warner type PDEs. This leads to novel explicit examples of quasi-Einstein structures on (non-compact) 2-manifolds and on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S^2 \times S^1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Quasi-Einstein Structures and Hitchin’s Equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年8月1日

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  • Alex Colling,
  • Maciej Dunajski

摘要

We prove (Theorem 1.1.) that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem obtained in [20] for the extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class. We also explore special cases of the quasi-Einstein equations related to integrability and the Hitchin equations, as well as to Einstein-Weyl structures and Kazdan-Warner type PDEs. This leads to novel explicit examples of quasi-Einstein structures on (non-compact) 2-manifolds and on \(S^2 \times S^1\) S 2 × S 1 .