In this paper we prove that the set of metrics conformal to the standard metric on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {S}}^{n}\backslash \{p_{1},\cdots ,p_{l}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="true">\</mo> </mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>l</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is locally compact in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{m,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> topology for any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, whenever the metrics have constant <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _{k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> curvature and the k-Dilational Pohozaev invariants have positive lower bound for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$k<n/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo><</mo> <mi>n</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Here the k-Dilational Pohozaev invariants come from the Kazdan-Warner type identity for the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _{k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> curvature, which is derived by Viaclovsky [<CitationRef CitationID="CR50">50</CitationRef>] and Han [<CitationRef CitationID="CR28">28</CitationRef>]. When <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$k=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, Pollack [<CitationRef CitationID="CR40">40</CitationRef>] proved the compactness results for the complete metrics of constant positive scalar curvature on <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2173_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {S}^{n}\backslash \{p_{1},\cdots ,p_{l}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="true">\</mo> </mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>l</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Compactness Theorem of Complete k-Curvature Manifolds with Isolated Singularities

期刊论文
发表日期：

2025年9月6日

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Wei Wei

摘要

In this paper we prove that the set of metrics conformal to the standard metric on ${\mathbb {S}}^{n}\backslash \{p_{1},\cdots ,p_{l}\}$ $S^{n} \ {p_{1}, \dots, p_{l}}$ is locally compact in $C^{m,\alpha }$ $C^{m, α}$ topology for any $m>0$ $m > 0$ , whenever the metrics have constant $\sigma _{k}$ $σ_{k}$ curvature and the k-Dilational Pohozaev invariants have positive lower bound for $k<n/2$ $k < n / 2$ . Here the k-Dilational Pohozaev invariants come from the Kazdan-Warner type identity for the $\sigma _{k}$ $σ_{k}$ curvature, which is derived by Viaclovsky [50] and Han [28]. When $k=1$ $k = 1$ , Pollack [40] proved the compactness results for the complete metrics of constant positive scalar curvature on $\mathbb {S}^{n}\backslash \{p_{1},\cdots ,p_{l}\}$ $S^{n} \ {p_{1}, \dots, p_{l}}$ .