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λ-Biharmonic hypersurfaces in the product space L^{m}\times \mathbb{R}

Chao Yang, Zhen Zhao

Abstract

In this paper, we study λ-biharmonic hypersurfaces in the product space L^{m}\times\mathbb{R}, where L^{m} is an Einstein space and \mathbb{R} is a real line. We prove that λ-biharmonic hypersurfaces with constant mean curvature in L^{m}\times\mathbb{R} are either minimal or vertical cylinders, and obtain some classification results for λ$-biharmonic hypersurfaces under various constraints. Furthermore, we investigate λ-biharmonic hypersurfaces in the product space L^{m}(c)\times\mathbb{R}, where L^{m}(c) is a space form with constant sectional curvature c, and categorize hypersurfaces that are either totally umbilical or semi-parallel.

λ-Biharmonic hypersurfaces in the product space L^{m}\times \mathbb{R}

Abstract

In this paper, we study λ-biharmonic hypersurfaces in the product space L^{m}\times\mathbb{R}, where L^{m} is an Einstein space and \mathbb{R} is a real line. We prove that λ-biharmonic hypersurfaces with constant mean curvature in L^{m}\times\mathbb{R} are either minimal or vertical cylinders, and obtain some classification results for λ$-biharmonic hypersurfaces under various constraints. Furthermore, we investigate λ-biharmonic hypersurfaces in the product space L^{m}(c)\times\mathbb{R}, where L^{m}(c) is a space form with constant sectional curvature c, and categorize hypersurfaces that are either totally umbilical or semi-parallel.
Paper Structure (5 sections, 78 equations)

This paper contains 5 sections, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Lemmas
  4. $\lambda$-biharmonic hypersurface in $L^{m}\times\mathbb{R}$
  5. $\lambda$-biharmonic hypersurface in $L^{m}(c)\times\mathbb{R}$