Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(M^n,g), n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mi>n</mi> </msup> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> be an asymptotically flat (AF) Riemannian manifold with nonnegative scalar curvature on which one of the Sobolev inequalities <Equation ID="Equ1"> <EquationNumber>0.1</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} \Big (\int _M|f|^pd\mu _g\Big )^{\frac{1}{p}}\le C \Big (\int _M|\nabla f|^qd\mu _g\Big )^{\frac{1}{q}}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <msub> <mo>∫</mo> <mi>M</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <mi>d</mi> <msub> <mi>μ</mi> <mi>g</mi> </msub> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </msup> <mo>≤</mo> <mi>C</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <msub> <mo>∫</mo> <mi>M</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> </msup> <mi>d</mi> <msub> <mi>μ</mi> <mi>g</mi> </msub> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f\in W^{1,q}(M), 1\le q<n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{p}=\frac{1}{q}-\frac{1}{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, is satisfied with C the optimal constant for this inequality on <InlineEquation ID="IEq4"> <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>, and suppose the positive mass theorem holds for AF manifolds, then (M, g) is isometric to <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>.

Sobolev Inequalities on Asymptotically Flat Manifolds with Nonnegative ADM Mass

期刊论文
发表日期：

2026年5月11日

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Yichao Li

摘要

Let $(M^n,g), n\ge 3$ $(M^{n}, g), n \geq 3$ be an asymptotically flat (AF) Riemannian manifold with nonnegative scalar curvature on which one of the Sobolev inequalities 0.1 $\begin{aligned} \Big (\int _M|f|^pd\mu _g\Big )^{\frac{1}{p}}\le C \Big (\int _M|\nabla f|^qd\mu _g\Big )^{\frac{1}{q}}, \end{aligned}$ $\begin{matrix} (\int_{M} {| f |}^{p} d μ_{g})^{\frac{1}{p}} \leq C (\int_{M} {| \nabla f |}^{q} d μ_{g})^{\frac{1}{q}}, \end{matrix}$ where $f\in W^{1,q}(M), 1\le q<n$ $f \in W^{1, q} (M), 1 \leq q < n$ and $\frac{1}{p}=\frac{1}{q}-\frac{1}{n}$ $\frac{1}{p} = \frac{1}{q} - \frac{1}{n}$ , is satisfied with C the optimal constant for this inequality on $\mathbb {R}^n$ $R^{n}$ , and suppose the positive mass theorem holds for AF manifolds, then (M, g) is isometric to $\mathbb {R}^n$ $R^{n}$ .