<p>In this paper, we show that any bilipschitz metric <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g\in W^{1,p}\bigcap C^0({\mathbb {R}}^n)(p>n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo>⋂</mo> <msup> <mi>C</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>></mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {R}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> which is smooth and has nonnegative scalar curvature outside a singular set of finite <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(n-\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional Minkowski content, for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda >\frac{3}{2}+\frac{n}{2p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mi>n</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, admits an approximation by smooth metrics with nonnegative scalar curvature. The approximation is provided by the Ricci-DeTurck flow.</p>

On nonnegative Scalar curvature outside a singular set

期刊论文
发表日期：

2025年8月27日

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Yuqiao Li

摘要

In this paper, we show that any bilipschitz metric $g\in W^{1,p}\bigcap C^0({\mathbb {R}}^n)(p>n)$ $g \in W^{1, p} ⋂ C^{0} (R^{n}) (p > n)$ on ${\mathbb {R}}^n$ $R^{n}$ which is smooth and has nonnegative scalar curvature outside a singular set of finite $(n-\lambda )$ $(n - λ)$ -dimensional Minkowski content, for $\lambda >\frac{3}{2}+\frac{n}{2p}$ $λ > \frac{3}{2} + \frac{n}{2 p}$ , admits an approximation by smooth metrics with nonnegative scalar curvature. The approximation is provided by the Ricci-DeTurck flow.