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<p>We construct all axi-symmetric non-gradient <i>m</i>-quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(m=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, as well as a family of new regular metrics with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m\ne 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> given in terms of hypergeometric functions. We also show that in the case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(m=-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.</p>

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New Quasi-Einstein Metrics on a Two-Sphere

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月16日

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  • Alex Colling,
  • Maciej Dunajski,
  • Hari Kunduri,
  • James Lucietti

摘要

We construct all axi-symmetric non-gradient m-quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to \(m=2\) m = 2 , as well as a family of new regular metrics with \(m\ne 2\) m ≠ 2 given in terms of hypergeometric functions. We also show that in the case \(m=-1\) m = - 1 with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.