伪黎曼乘积空间 N_p^m(c)× ℝ中的λ-双调和超曲面

λ-Biharmonic Hypersurfaces in Pseudo-Riemannian Product Space N_p^m(c)× ℝ

摘要: 本文主要研究伪黎曼乘积空间N_p^m(c)× ℝ中的λ-双调和超曲面，给出超曲面是λ-双调和的等价方程，证得N_p^m(c)× ℝ中具有常平均曲率且形状算子可对角化的λ-双调和(ε_m+1λ≥0)超曲面要么是极小的，要么是一个直柱体。利用该结论,在角度函数为常数的假设下，对N_p^m(c)× ℝ中的Einstein 型 λ-双调和超曲面进行分类。特别地，我们讨论了(ℍ^m(c) × ℝ,g^N- dt²)中至多具有两个不同主曲率的λ-双调和类空超曲面(M^m,g),在角度函数是常数且双曲角a≠0的假设下证得超曲面M^m要么是极小的,要么是一个直柱体。

Abstract: In this paper, we study the λ-biharmonic hypersurfaces in the pseudo-Riemannian product space N_p^m(c)× ℝ, and derive λ-biharmonic equation. It is shown that the λ-biharmonic hypersurfaces (ε_m+1λ≥0) with constant mean curvature and shape op- erator can be diagonalizable are either minimal or a vertical cylinder. Utilizing this result, the paper classifies Einstein-type λ-biharmonic hypersurfaces in N_p^m(c)× ℝ under the assumption of a constant angle function. In particular, it classifies at most two distinct principal curvatures of λ-biharmonic space-like hypersurfaces (M^m,g) in (ℍ^m(c) × ℝ,g^N- dt²) , and proves that under the assumption of a constant angle and hyperbolic angle a≠0, the hypersurface M^m is either minimal or a vertical cylinder.