Optimal Regularity of Constant Graphs in Hyperbolic Space

TL;DR

This paper investigates the boundary regularity of constant curvature hypersurfaces in hyperbolic space with prescribed asymptotic boundary, establishing optimal regularity results and conditions for smoothness.

Contribution

It provides the first comprehensive boundary regularity analysis for such hypersurfaces, deriving optimal regularity and smoothness criteria.

Findings

Derived boundary expansions of solutions

Established optimal regularity results

Identified conditions for solution smoothness

Abstract

Inspired by [6, 7], we study the boundary regularity of constant curvature hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$, which have prescribed asymptotic boundary at infinity. Through constructing the boundary expansions of the solutions, we derive the optimal regularity of the solutions. Moreover, we obtain an equivalent condition that guarantees the smoothness of the solutions.