Abstract
In this paper we consider codimension two marginally trapped submanifolds in the family of general Robertson–Walker spacetimes. In particular, we derive some rigidity results for this type of submanifolds which guarantee that, under appropriate hypothesis, the only ones are those contained in slices. We also derive some interesting non-existence results for weakly trapped submanifolds. In particular, we give applications to some cases of physical relevance such as the Einstein-de Sitter spacetime and certain open regions of de Sitter spacetime, including the so called steady state spacetime. Our results will be an application of the (finite) maximum principle for closed manifolds and, more generally, of the weak maximum principle for stochastically complete manifolds.
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Acknowledgements
The authors would like to thank the anonymous referee for his/her valuable suggestions and corrections which contributed very much to improve this paper. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. This work was partially supported by MINECO/FEDER project reference MTM2015-65430-P and Fundación Séneca project reference 19901/GERM/15, Spain. L. J. Alías was also partially supported by Ciência sem Fronteiras, Programa PVE, Project A012/2013, CAPES, Brazil. V. L. Cánovas was also supported by research Grant 19783/FPI/15 from Fundación Séneca, Murcia, Spain. A. G. Colares was partially supported by CNPq, Brazil.
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Alías, L.J., Cánovas, V.L. & Colares, A.G. Marginally trapped submanifolds in generalized Robertson–Walker spacetimes. Gen Relativ Gravit 49, 23 (2017). https://doi.org/10.1007/s10714-017-2188-9
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DOI: https://doi.org/10.1007/s10714-017-2188-9
Keywords
- General Robertson–Walker spacetimes
- Marginally trapped submanifolds
- Mean curvature
- Maximum principle
- Weak maximum principle