Abstract
We revisit the problem of studying the geometry of horospheres of the hyperbolic space \(\mathbb {H}^{n+1}\), with the purpose of characterizing them under certain appropriate constraints in the behavior of their higher order mean curvatures. In particular, we obtain a gap type result concerning the scalar curvature of complete two-sided hypersurfaces immersed in \(\mathbb {H}^{n+1}\). Furthermore, we establish a lower estimate for the index of minimum relative nullity of r-minimal (\(2\le r\le n-1\)) hypersurfaces of \(\mathbb {H}^{n+1}\) and we also get a nonexistence result for 1-minimal hypersurfaces in the closed horoball determined by a horosphere of \(\mathbb H^{n+1}\). Our approach is based on a suitable version of the generalized maximum principle of Omori–Yau for trace-type operators defined on a complete Riemannian manifold with sectional curvature bounded from below.
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The second and third authors are partially supported by CNPq, Brazil, Grants 301970/2019-0 and 311224/2018-0, respectively.
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Communicated by P. Chrusciel.
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Colares, A.G., de Lima, H.F. & Velásquez, M.A.L. Revisiting the geometry of horospheres of the hyperbolic space. Monatsh Math 199, 771–784 (2022). https://doi.org/10.1007/s00605-022-01758-2
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DOI: https://doi.org/10.1007/s00605-022-01758-2
Keywords
- Hyperbolic space
- Complete two-sided hypersurfaces
- Horospheres
- Higher order mean curvatures
- r-minimal hypersurfaces
- Index of minimum relative nullity