本文主要研究伪黎曼乘积空间Npm(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Npm(c)× ℝ中具有常平均曲率且形状算子可对角化的λ-双调和(εm+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Npm(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(ℍm(c) × ℝ,gN- dt2)中至多具有两个不同主曲率的λ-双调和类空超曲面(Mm,g),在角度函数是常数且双曲角a≠0的假设下证得超曲面Mm要么是极小的,要么是一个直柱体。In this paper, we study the λ-biharmonic hypersurfaces in the pseudo-Riemannian product space Npm(c)× ℝ, and derive λ-biharmonic equation. It is shown that the λ-biharmonic hypersurfaces (εm+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 Npm(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 (Mm,g) in (ℍm(c) × ℝ,gN- dt2) , and proves that under the assumption of a constant angle and hyperbolic angle a≠0, the hypersurface Mm is either minimal or a vertical cylinder.
In this paper,we prove the boundedness of the singular integral operators on product spaces with mixed norms and obtain the endpoint weak-type estimates.
