Junhong Dong, Ximin Liu: Uniqueness of complete spacelike hypersurface in Lorentzian warped products, p. 33-49

Abstract:

In this paper, we apply several forms of generalized maximum principles to study the uniqueness of complete spacelike hypersurfaces immersed in Lorentzian warped products. First, we consider the cases of ambient space with vanish $f'$, then obtain some uniqueness results of constant $k$-th mean curvature. Afterwards, we obtain the sign relationship between the support function with the derivative of warping function. By using this result, under some suitable restriction on the higher order mean curvature, we establish the uniqueness results of Lorentzian warped product $-\mathbb{R}\times_{f} M^{n}$ with non-vanish $f'$. Furthermore, applications to such spaces are given.

Key Words: Lorentzian warped product, warping function, complete hypersurface, higher order mean curvature, spacelike slice

2010 Mathematics Subject Classification: Primary 53C42,
Secondary 53B30, 53C50, 53Z05