This paper investigated $ \lambda $-biharmonic hypersurfaces in $ {L}^{m}\times \mathbb{R} $, where $ {L}^{m} $ represents an Einstein space, and $ \mathbb{R} $ denotes the real line. We demonstrated that such hypersurfaces with some curvature conditions are either of two types: minimal or vertical cylinders over $ \lambda $-biharmonic hypersurfaces in $ {L}^{m} $. Particularly, when the Einstein space $ {L}^{m} $ has constant sectional curvature, we classify $ \lambda $-biharmonic hypersurfaces as totally umbilical or semi-parallel.
Citation: Jiarui Chen, Zhen Zhao, Chao Yang. On $ {{\rm{ \mathsf{ λ}}} } $-biharmonic hypersurfaces in $ {{L}}^{{m}}\times \mathbb{R} $[J]. Electronic Research Archive, 2025, 33(8): 4723-4739. doi: 10.3934/era.2025212
This paper investigated $ \lambda $-biharmonic hypersurfaces in $ {L}^{m}\times \mathbb{R} $, where $ {L}^{m} $ represents an Einstein space, and $ \mathbb{R} $ denotes the real line. We demonstrated that such hypersurfaces with some curvature conditions are either of two types: minimal or vertical cylinders over $ \lambda $-biharmonic hypersurfaces in $ {L}^{m} $. Particularly, when the Einstein space $ {L}^{m} $ has constant sectional curvature, we classify $ \lambda $-biharmonic hypersurfaces as totally umbilical or semi-parallel.
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Jiarui Chen, Zhen Zhao, Chao Yang. On $ {{\rm{ \mathsf{ λ}}} } $-biharmonic hypersurfaces in $ {{L}}^{{m}}\times \mathbb{R} $[J]. Electronic Research Archive, 2025, 33(8): 4723-4739. doi: 10.3934/era.2025212
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