Asymptotic Plateau problem via equidistant hyperplanes

TL;DR

This paper introduces a new method for solving the asymptotic Plateau problem in hyperbolic space by constructing convex hypersurfaces as geodesic graphs over bounded domains, using equidistant hyperplanes for approximation.

Contribution

It provides an alternative approach to the asymptotic Plateau problem by constructing solutions via equidistant hyperplanes, expanding the types of solutions beyond classical ones.

Findings

Existence of convex hypersurfaces with prescribed asymptotic boundary.

Construction method using geodesic graphs and equidistant hyperplanes.

Solutions may differ from classical solutions when uniqueness fails.

Abstract

We show the existence of a complete, strictly locally convex hypersurface within $\mathbb{H}^{n+1}$ that adheres to a curvature equation applicable to a broad range of curvature functions. This hypersurface possesses a prescribed asymptotic boundary at infinity and takes the form of a geodesic graph over a smooth bounded domain $\Omega$ at infinity. It is approximated by the shape of geodesic graphs whose boundaries rest upon equidistant hyperplanes. Through this procedure, we establish an alternative method for constructing solutions to the asymptotic Plateau problem. The resulting solutions may differ from the classical ones, particularly in cases where uniqueness cannot be assured.