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<p>In this paper we prove <b>localised weighted</b> curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed <i>n</i>-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> sense for some <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p&gt;2,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> then the estimates imply a uniform bound on the spatial <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.</p>

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Integral curvature estimates for solutions to Ricci flow with Lp bounded scalar curvature

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  • 发表日期：

    2025年11月15日

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  • Jiawei Liu,
  • Miles Simon

摘要

In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed n-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial \(L^p\) L p sense for some \(p>2,\) p > 2 , then the estimates imply a uniform bound on the spatial \(L^2\) L 2 norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.