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<p>In this paper, we study scalar curvature rigidity of non-smooth metrics on smooth manifolds with non-positive Yamabe invariant. We prove that if the scalar curvature is not less than the Yamabe invariant in the distributional sense, then the manifold must be isometric to an Einstein manifold. This result extends Theorem 1.4 in Jiang, Sheng and Zhang [<CitationRef CitationID="CR27">27</CitationRef>], from a special case where the manifolds have zero Yamabe invariant to general cases where the manifolds have non-positive Yamabe invariant.</p>

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Distributional scalar curvature and Einstein metrics

  • 期刊论文

  • 发表日期：

    2025年6月26日

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  • Huai-yu Zhang,
  • Jiang-wei Zhang

摘要

In this paper, we study scalar curvature rigidity of non-smooth metrics on smooth manifolds with non-positive Yamabe invariant. We prove that if the scalar curvature is not less than the Yamabe invariant in the distributional sense, then the manifold must be isometric to an Einstein manifold. This result extends Theorem 1.4 in Jiang, Sheng and Zhang [27], from a special case where the manifolds have zero Yamabe invariant to general cases where the manifolds have non-positive Yamabe invariant.