We show that any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> Riemannian metric g on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> that is smooth with nonnegative scalar curvature away from a singular set of finite <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(n-\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional Minkowski content, for some <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha >2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that g is sufficiently close in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> to the Euclidean metric. The approximation is given by time slices of the Ricci–DeTurck flow, which converge locally in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$C^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> to g away from the singular set. We also identify conditions under which a smooth Ricci–DeTurck flow starting from a <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.

Smoothing $L^\infty $ Riemannian metrics with nonnegative scalar curvature outside of a singular set

期刊论文
发表日期：

2026年5月11日

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Paula Burkhardt-Guim

摘要

We show that any $L^\infty $ $L^{\infty}$ Riemannian metric g on $\mathbb {R}^n$ $R^{n}$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-\alpha )$ $(n - α)$ -dimensional Minkowski content, for some $\alpha >2$ $α > 2$ , admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that g is sufficiently close in $L^\infty $ $L^{\infty}$ to the Euclidean metric. The approximation is given by time slices of the Ricci–DeTurck flow, which converge locally in $C^\infty $ $C^{\infty}$ to g away from the singular set. We also identify conditions under which a smooth Ricci–DeTurck flow starting from a $L^\infty $ $L^{\infty}$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.

Smoothing \(L^\infty \) Riemannian metrics with nonnegative scalar curvature outside of a singular set

摘要