Abstract
Our aim in this article is to study the rigidity of complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type \(\mathbb {R}_1\times \mathbb {P}^n_f\), whose Bakry–Émery-Ricci tensor of the fiber \(\mathbb {P}^n\) is nonnegative and the Hessian of the weighted function f is bounded from below. In this setting, supposing that the weighted mean curvature \(H_f\) is constant and assuming appropriated constraints on the norm of the gradient of the height function, we prove that such a hypersurface must be a slice of the ambient space. Applications to entire spacelike graphs construct over \(\mathbb {P}^n\) are also given. Our approach is based on a suitable formula for the drift Laplacian of an angle function naturally attached to a spacelike hypersurface jointly with a weak version of the Omori–Yau’s generalized maximum principle.
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Acknowledgments
The first author is partially supported by CNPq, Brazil, Grant 300769/2012-1. The second author is partially supported by CAPES, Brazil. The authors would like to thank the referee for giving several valuable suggestions which improved the paper very much.
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de Lima, H.F., Oliveira, A.M.S. & Santos, M.S. Rigidity of complete spacelike hypersurfaces with constant weighted mean curvature. Beitr Algebra Geom 57, 623–635 (2016). https://doi.org/10.1007/s13366-015-0253-7
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DOI: https://doi.org/10.1007/s13366-015-0253-7
Keywords
- Weighted Lorentzian product spaces
- Bakry–Émery-Ricci tensor
- Drifting Laplacian
- Complete spacelike hypersurfaces
- Entire spacelike graphs
- f-Mean curvature