<p>In this paper, we prove that a closed minimally immersed hypersurface <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2129_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$M^4\subset {\mathbb {S}}^5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mn>4</mn> </msup> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>5</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with constant <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2129_Article_IEq7.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$S:=\sum \limits _{i=1}^4\lambda _i^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>:</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </munderover> <msubsup> <mi>λ</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2129_Article_IEq8.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$A_3:=\sum \limits _{i=1}^4\lambda _i^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mn>3</mn> </msub> <mo>:</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </munderover> <msubsup> <mi>λ</mi> <mi>i</mi> <mn>3</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> whose scalar curvature <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2129_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$R_M$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>M</mi> </msub> </math></EquationSource> </InlineEquation> is nonnegative must be isoparametric. Moreover, <i>S</i> can only be 0, 4,  and 12. That is <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2129_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$M^4$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> is either an equatorial 4-sphere, a clifford torus, or a Cartan’s minimal hypersurface.</p>

Closed Minimal Hypersurfaces in ${\mathbb {S}}^5$ with Constant S and $A_3$

期刊论文
发表日期：

2025年7月30日

查看全文

Joel Spruck,
LIng XIao

摘要

In this paper, we prove that a closed minimally immersed hypersurface $M^4\subset {\mathbb {S}}^5$ $M^{4} \subset S^{5}$ with constant $S:=\sum \limits _{i=1}^4\lambda _i^2$ $S : = \sum_{i = 1}^{4} λ_{i}^{2}$ and $A_3:=\sum \limits _{i=1}^4\lambda _i^3$ $A_{3} : = \sum_{i = 1}^{4} λ_{i}^{3}$ whose scalar curvature $R_M$ $R_{M}$ is nonnegative must be isoparametric. Moreover, S can only be 0, 4, and 12. That is $M^4$ $M^{4}$ is either an equatorial 4-sphere, a clifford torus, or a Cartan’s minimal hypersurface.

Closed Minimal Hypersurfaces in \({\mathbb {S}}^5\) with Constant S and \(A_3\)

摘要