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Weakly horospherically convex hypersurfaces in hyperbolic space
- Source :
- Annals of Global Analysis and Geometry, vol 52, iss 2, Bonini, Vincent; Qing, Jie; & Zhu, Jingyong. (2019). Weakly Horospherically Convex Hypersurfaces in Hyperbolic Space. UC Santa Cruz: Retrieved from: http://www.escholarship.org/uc/item/8df3j0tn
- Publication Year :
- 2017
- Publisher :
- eScholarship, University of California, 2017.
-
Abstract
- In [2], the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $\phi:M^n \to \mathbb{H}^{n+1}$ and a class of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Some of the key aspects of the correspondence and its consequences have dimensional restrictions $n\geq3$ due to the reliance on an analytic proposition from [5] concerning the asymptotic behavior of conformal factors of conformal metrics on domains of $\mathbb{S}^n$. In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of [2] to all dimensions $n\geq2$ in a unified way. In the case of a single point boundary $\partial_{\infty}\phi(M)=\{x\} \subset \mathbb{S}^n$, we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in [2], we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from [2] to the case of surfaces in $\mathbb{H}^{3}$.<br />Comment: Welcome comments!
- Subjects :
- Mathematics - Differential Geometry
Class (set theory)
Pure mathematics
Lemma (mathematics)
Delaunay triangulation
Hyperbolic space
General Mathematics
010102 general mathematics
Regular polygon
Boundary (topology)
Conformal map
Type (model theory)
01 natural sciences
Pure Mathematics
math.DG
Differential Geometry (math.DG)
0103 physical sciences
FOS: Mathematics
010307 mathematical physics
Geometry and Topology
0101 mathematics
Analysis
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Annals of Global Analysis and Geometry, vol 52, iss 2, Bonini, Vincent; Qing, Jie; & Zhu, Jingyong. (2019). Weakly Horospherically Convex Hypersurfaces in Hyperbolic Space. UC Santa Cruz: Retrieved from: http://www.escholarship.org/uc/item/8df3j0tn
- Accession number :
- edsair.doi.dedup.....72fc7f6152f2c21051822a45ca47e9fc