Weakly Horospherically Convex Hypersurfaces in Hyperbolic Space

TL;DR

This paper extends the global correspondence between weakly horospherically convex hypersurfaces in hyperbolic space and conformal metrics on spheres to all dimensions, providing new results on embeddedness, Bernstein theorems, and classifications of such hypersurfaces.

Contribution

It introduces a new lemma on asymptotic behavior, enabling the extension of key theorems to all dimensions n≥2, unifying previous dimension restrictions.

Findings

Extended the global correspondence to all dimensions n≥2.

Proved a new Bernstein type theorem for hypersurfaces.

Extended Liouville and Delaunay theorems to surfaces in hyperbolic 3-space.

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 $\mathbb{H}^{3}$.