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



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URL: https://digitalcommons.calpoly.edu/math_fac/117