<p>In this paper, we study <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1705_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-biharmonic hypersurfaces in a Euclidean space. Using a Murnaghan–Nakayama type formula, we prove that any <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1705_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-biharmonic hypersurface of the 6-dimensional Euclidean space is either minimal or it has mean curvature and constant scalar curvature if <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1705_Article_IEq9.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and it is minimal if <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1705_Article_IEq10.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \le 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. This extends a recent work (Fu et al. in Adv Math 383:107697, 2021) [<CitationRef CitationID="CR16">16</CitationRef>] and give further progress on a problem proposed by Chen in 1991. We also obtain an estimate of the mean curvature of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1705_Article_IEq11.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-biharmonic hypersurfaces in a Euclidean space.</p>

On $\alpha $-biharmonic hypersurfaces in $\mathbb {R}^6$

期刊论文
发表日期：

2025年2月21日

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Dan Yang,
Xin Zhan

摘要

In this paper, we study $\alpha $ $α$ -biharmonic hypersurfaces in a Euclidean space. Using a Murnaghan–Nakayama type formula, we prove that any $\alpha $ $α$ -biharmonic hypersurface of the 6-dimensional Euclidean space is either minimal or it has mean curvature and constant scalar curvature if $\alpha >0$ $α > 0$ , and it is minimal if $\alpha \le 0$ $α \leq 0$ . This extends a recent work (Fu et al. in Adv Math 383:107697, 2021) [16] and give further progress on a problem proposed by Chen in 1991. We also obtain an estimate of the mean curvature of $\alpha $ $α$ -biharmonic hypersurfaces in a Euclidean space.

On \(\alpha \)-biharmonic hypersurfaces in \(\mathbb {R}^6\)

摘要