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<p>Let (<i>M</i>,&#xa0;<i>g</i>) be a four-dimensional closed connected oriented (possibly non-spin) Riemannian manifold whose scalar curvature is bounded below by 12. We prove that, if <i>f</i> is a smooth distance non-increasing map of non-zero degree from (<i>M</i>,&#xa0;<i>g</i>) to the unit four-sphere, then <i>f</i> is an isometry. This removes the spin condition in Llarull’s scalar curvature rigidity theorem for spheres in dimension four.</p>

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Scalar curvature rigidity of the four-dimensional sphere

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月18日

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  • Simone Cecchini,
  • Jinmin Wang,
  • Zhizhang Xie,
  • Bo Zhu

摘要

Let (M, g) be a four-dimensional closed connected oriented (possibly non-spin) Riemannian manifold whose scalar curvature is bounded below by 12. We prove that, if f is a smooth distance non-increasing map of non-zero degree from (M, g) to the unit four-sphere, then f is an isometry. This removes the spin condition in Llarull’s scalar curvature rigidity theorem for spheres in dimension four.