Hypersurfaces of constant curvature in Hyperbolic space

TL;DR

This paper proves the existence and uniqueness of convex hypersurfaces with prescribed curvature and boundary at infinity in hyperbolic space, showing they form foliations of the space's complement.

Contribution

It establishes existence, uniqueness, and foliation properties for hypersurfaces with general curvature functions in hyperbolic space.

Findings

Existence of solutions for a broad class of curvature functions.

Uniqueness for a specific subclass of curvature functions.

Hypersurfaces foliate the complement of the hyperbolic convex hull of the boundary.

Abstract

We show that for a very general and natural class of curvature functions, the problem of finding a complete strictly convex hypersurface satisfying f({\kappa}) = {\sigma} over (0,1) with a prescribed asymptotic boundary {\Gamma} at infinity has at least one solution which is a "vertical graph" over the interior (or the exterior) of {\Gamma}. There is uniqueness for a certain subclass of these curvature functions and as {\sigma} varies between 0 and 1, these hypersurfaces foliate the two components of the complement of the hyperbolic convex hull of {\Gamma}.