<p>In this paper, we prove that for every compact non-degenerate <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2124_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> star-shaped hypersurface <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2124_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2124_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^{8}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>8</mn> </msup> </math></EquationSource> </InlineEquation> which carries no prime closed characteristics with Maslov-type index <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2124_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist at least four prime closed characteristics on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2124_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>.</p>

Four closed characteristics on compact star-shaped hypersurfaces in $\mathbb {R}^{8}$

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发表日期：

2025年7月26日

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Huagui Duan,
Dong Xie

摘要

In this paper, we prove that for every compact non-degenerate $C^3$ $C^{3}$ star-shaped hypersurface $\Sigma $ $Σ$ in $\mathbb {R}^{8}$ $R^{8}$ which carries no prime closed characteristics with Maslov-type index $-1$ $- 1$ , there exist at least four prime closed characteristics on $\Sigma $ $Σ$ .

Four closed characteristics on compact star-shaped hypersurfaces in \(\mathbb {R}^{8}\)

摘要