@article{2025arXiv250805349T,author={{Trebeschi}, Enrico},title={{Generalized convexity and quantitative estimates for constant mean curvature spacelike hypersurfaces in Anti-de Sitter space}},journal={arXiv e-prints},year={2025},keywords={preprint},eprint={2508.05349},doi={10.48550/arXiv.2508.05349},archiveprefix={arXiv},primaryclass={math.DG},adsurl={https://ui.adsabs.harvard.edu/abs/2025arXiv250805349T},adsnote={Provided by the SAO/NASA Astrophysics Data System}}
2024
Preprint
On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces
We prove that the scalar curvature is a rigid invariant for complete maximal spacelike p-submanifolds in the pseudo-hyperbolic space Hp,q. We characterize the p-submanifolds achieving the bound and study the hyperbolicity of maximal p-submanifolds of Hp,q. In codimension q=1, we deduce a rigidity result concerning the Ricci curvature of maximal hypersurfaces in anti-de Sitter spaces.
@article{2024arXiv241110352M,author={{Moriani}, Alex and {Trebeschi}, Enrico},title={{On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces}},journal={arXiv e-prints},year={2024},keywords={preprint},eprint={2411.10352},adsurl={https://ui.adsabs.harvard.edu/abs/2024arXiv241110352M},adsnote={Provided by the SAO/NASA Astrophysics Data System},}
IMRN
Constant mean curvature hypersurfaces in Anti-de Sitter space
We study entire spacelike constant mean curvature hypersurfaces in Anti-de Sitter space of any dimension. First, we give a classification result with respect to their asymptotic boundary, namely we show that every admissible sphere \(Λ\)is the boundary of a unique such hypersurface, for any given value \(H\)of the mean curvature. We also demonstrate that, as \(H\)varies in \(\mathbbR\), these hypersurfaces analytically foliate the invisible domain of \(Λ\). Finally, we extend Cheng-Yau Theorem to the Anti-de Sitter space, which establishes the completeness of any entire constant mean curvature hypersurface.
@article{trebeschi2024constant,title={Constant mean curvature hypersurfaces in Anti-de Sitter space},author={Trebeschi, Enrico},journal={International Mathematics Research Notices},volume={2024},number={9},pages={8026--8066},year={2024},publisher={Oxford University Press},keywords={published},doi={10.1093/imrn/rnae032},url={https://doi.org/10.1093/imrn/rnae032},adsurl={https://ui.adsabs.harvard.edu/abs/2023arXiv230812167T},adsnote={Provided by the SAO/NASA Astrophysics Data System},}
In this note we develop a half-space model for the pseudo-hyperbolic space \(\mathbbH^p,q\), for any \(p,q\)with \(p≥1\). This half-space model embeds isometrically onto the complement of a degenerate totally geodesic hyperplane in \(\mathbbH^p,q\). We describe the geodesics, the totally geodesic submanifolds, the horospheres, the isometry group in the half-space model, and we explain how to interpret the boundary at infinity in this setting.
@inproceedings{seppi2021half,title={The half-space model of pseudo-hyperbolic space},author={Seppi, Andrea and Trebeschi, Enrico},booktitle={International Meeting on Lorentzian Geometry},pages={285--313},year={2021},organization={Springer},keywords={published},doi={10.1007/978-3-031-05379-5_17},url={https://link.springer.com/chapter/10.1007/978-3-031-05379-5_17},adsurl={https://ui.adsabs.harvard.edu/abs/2021arXiv210611122S},adsnote={Provided by the SAO/NASA Astrophysics Data System},}