Weakly horospherically convex hypersurfaces in hyperbolic space

Abstract

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $\phi :M^n \rightarrow \mathbb {H}^{n+1}$ 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 $n\ge 3$ 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 $\mathbb {S}^n$. 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 Bonini et al. (2015) to all dimensions $n\ge 2$ 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 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 $\mathbb {H}^{3}$.