<p>By using the Yamabe flow, we prove that if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2971_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$(M^n,g)$</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> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2971_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, is an <i>n</i>-dimensional locally conformally flat complete Riemannian manifold satisfying <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2971_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">$Rc\ge \epsilon Rg>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>c</mi> <mo>≥</mo> <mi>ϵ</mi> <mi>R</mi> <mi>g</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2971_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a uniformly constant, then <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2971_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$M^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> must be compact. Our result shows that Hamilton’s pinching conjecture also holds for higher dimensional case if we assume additionally the metric is locally conformally flat.</p>

Yamabe flow and locally conformally flat manifolds with positive pinched Ricci curvature

期刊论文
发表日期：

2025年3月13日

查看全文

Liang Cheng

摘要

By using the Yamabe flow, we prove that if $(M^n,g)$ $(M^{n}, g)$ , $n\ge 3$ $n \geq 3$ , is an n-dimensional locally conformally flat complete Riemannian manifold satisfying $Rc\ge \epsilon Rg>0$ $R c \geq ϵ R g > 0$ , where $\epsilon >0$ $ϵ > 0$ is a uniformly constant, then $M^n$ $M^{n}$ must be compact. Our result shows that Hamilton’s pinching conjecture also holds for higher dimensional case if we assume additionally the metric is locally conformally flat.