In this paper we study n-dimensional Ricci flows <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(M^n,g(t))_{t\in [0,T)},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mi>n</mi> </msup> <mo>,</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T< \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> is a potentially singular time, and for which the spatial <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> norm, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p>\frac{n}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, of the scalar curvature is uniformly bounded on [0, T). In the case that M is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper [<CitationRef CitationID="CR15">15</CitationRef>], as well as non-inflating estimates (already known due to works of Bamler [<CitationRef CitationID="CR5">5</CitationRef>]), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, then the solution convergences to an orbifold as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t \rightarrow T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and that the flow can be extended using the Orbifold Ricci flow to the time interval <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ [0,T+\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\sigma >0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We also prove local versions of many of the results mentioned above.

Volume estimates and convergence results for solutions to Ricci flow with $L^{p}$ bounded scalar curvature

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2026年4月13日

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Jiawei Liu,
Miles Simon

摘要

In this paper we study n-dimensional Ricci flows $(M^n,g(t))_{t\in [0,T)},$ ${(M^{n}, g (t))}_{t \in [0, T)},$ where $T< \infty $ $T < \infty$ is a potentially singular time, and for which the spatial $L^p$ $L^{p}$ norm, $p>\frac{n}{2}$ $p > \frac{n}{2}$ , of the scalar curvature is uniformly bounded on [0, T). In the case that M is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper [15], as well as non-inflating estimates (already known due to works of Bamler [5]), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to $n=4$ $n = 4$ , then the solution convergences to an orbifold as $t \rightarrow T$ $t \to T$ and that the flow can be extended using the Orbifold Ricci flow to the time interval $ [0,T+\sigma )$ $[0, T + σ)$ for some $\sigma >0.$ $σ > 0 .$ We also prove local versions of many of the results mentioned above.

Volume estimates and convergence results for solutions to Ricci flow with \(L^{p}\) bounded scalar curvature

摘要