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<p>In this paper, we prove that four-dimensional hypersurface <i>M</i><Stack> <sub><i>r</i></sub> <sup>4</sup> </Stack> with proper mean curvature vector field (i.e., <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta{\vec H}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi mathvariant="normal">Δ</mi> <mrow> <mrow> <mover> <mi>H</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </math></EquationSource> </InlineEquation> is proportional to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\vec H}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mover> <mi>H</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </math></EquationSource> </InlineEquation>) in pseudo-Riemannian space form <i>N</i><Stack> <sub><i>s</i></sub> <sup>5</sup> </Stack>(<i>c</i>) has constant mean curvature, and give the value or range of this constant. As an application, we obtain that biharmonic hypersurfaces in <i>N</i><Stack> <sub><i>s</i></sub> <sup>5</sup> </Stack>(<i>c</i>) are minimal in some specific case.</p>

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Four Dimensional Hypersurfaces with Proper Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

  • 期刊论文

  • 发表日期：

    2026年1月15日

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  • Chao Yang,
  • Jiancheng Liu,
  • Li Du

摘要

In this paper, we prove that four-dimensional hypersurface M r 4 with proper mean curvature vector field (i.e., \(\Delta{\vec H}\) Δ H → is proportional to \({\vec H}\) H → ) in pseudo-Riemannian space form N s 5 (c) has constant mean curvature, and give the value or range of this constant. As an application, we obtain that biharmonic hypersurfaces in N s 5 (c) are minimal in some specific case.