Recommended Citation
Postprint version. Published in Annals of Global Analysis and Geometry, Volume 52, Issue 2, September 1, 2017, pages 201-212.
The definitive version is available at https://doi.org/10.1007/s10455-017-9554-4.
Abstract
In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces ϕ:Mn→Hn+1 and a class of conformal metrics on domains of the round sphere Sn . Some of the key aspects of the correspondence and its consequences have dimensional restrictions n≥3 due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of Sn . 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 Bonini et al. (2015) to all dimensions n≥2 in a unified way. In the case of a single point boundary ∂∞ϕ(M)={x}⊂Sn , we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in H3
Disciplines
Mathematics
Copyright
Copyright © 2017 Springer.
Number of Pages
12
URL: https://digitalcommons.calpoly.edu/math_fac/117