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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$M$</EquationSource> </InlineEquation> be a closed 3-dimensional Riemannian manifold with positive scalar curvature, <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>≥</mo> <mn>6</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{g} \geq 6$</EquationSource> </InlineEquation>. We show that <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$M$</EquationSource> </InlineEquation> contains a non-trivial closed geodesic of length less than <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mn>22</mn> <mo>,</mo> <mn>500</mn> </math></EquationSource> <EquationSource Format="TEX">$22{,}500$</EquationSource> </InlineEquation>. This confirms a conjecture of M. Gromov in dimension 3.</p>

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Length of a Closed Geodesic in 3-Manifolds of Positive Scalar Curvature

  • 期刊论文

  • 发表日期：

    2026年5月28日

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  • Yevgeny Liokumovich,
  • Davi Maximo,
  • Regina Rotman

摘要

Let M $M$ be a closed 3-dimensional Riemannian manifold with positive scalar curvature, R g ≥ 6 $R_{g} \geq 6$ . We show that M $M$ contains a non-trivial closed geodesic of length less than 22 , 500 $22{,}500$ . This confirms a conjecture of M. Gromov in dimension 3.