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<p>We sharpen a gap theorem of Chan and Lee (Gap theorem on manifolds with small curvature concentration. <a href="http://arxiv.org/abs/2312.07845v1">arXiv:2312.07845v1</a>) for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called “curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than 1/<i>t</i> curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.</p>

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Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Adam Martens

摘要

We sharpen a gap theorem of Chan and Lee (Gap theorem on manifolds with small curvature concentration. arXiv:2312.07845v1) for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called “curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than 1/t curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.