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<p>We show that any complete minimal hypersurface in the five-dimensional hyperbolic space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {H}^5\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>5</mn> </msup> </math></EquationSource> </InlineEquation>, endowed with constant scalar curvature and vanishing Gauss-Kronecker curvature, must be totally geodesic. Cheng and Peng [<CitationRef CitationID="CR3">3</CitationRef>] recently conjectured that any complete minimal hypersurface with constant scalar curvature in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {H}^4\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> is totally geodesic. Our result partially confirms this conjecture in the five dimensional setting.</p>

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Complete Minimal Hypersurfaces in \(\mathbb {H}^5\) with Constant Scalar Curvature and Zero Gauss-Kronecker Curvature

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Qing Cui,
  • Boyuan Zhang

摘要

We show that any complete minimal hypersurface in the five-dimensional hyperbolic space \(\mathbb {H}^5\) H 5 , endowed with constant scalar curvature and vanishing Gauss-Kronecker curvature, must be totally geodesic. Cheng and Peng [3] recently conjectured that any complete minimal hypersurface with constant scalar curvature in \(\mathbb {H}^4\) H 4 is totally geodesic. Our result partially confirms this conjecture in the five dimensional setting.