Spacelike Graphs with Parallel Mean Curvature in Pseudo-Riemannian Product Manifolds |
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Authors: | Zicheng ZHAO |
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Institution: | (1) School of Mathematical Sciences, Fudan University, Shanghai, 200433, China |
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Abstract: | The author introduces the w-function defined on the considered spacelike graph M. Under the growth conditions w = o(log z) and w = o(r), two Bernstein type theorems for M in ℝ
m
n+m
are got, where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively. As the ambient space is a curved pseudo-Riemannian product of two Riemannian manifolds
(Σ1, g
1) and (Σ2, g
2) of dimensions n and m, a Bernstein type result for n = 2 under some curvature conditions on Σ1 and Σ2 and the growth condition w = o(r) is also got. As more general cases, under some curvature conditions on the ambient space and the growth condition w = o(r) or w = 0( ?r )w = 0\left( {\sqrt r } \right), the author concludes that if M has parallel mean curvature, then M is maximal. |
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Keywords: | Product manifold Spacelike graph Parallel mean curvature Maximal Bernstein |
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