Boundary regularity of minimal graphs in the hyperbolic space

TL;DR

This paper investigates the boundary regularity of minimal graphs in hyperbolic space, focusing on the odd-dimensional case, and introduces a new approach using the logarithm of the distance to the boundary to understand regularity obstructions.

Contribution

It provides a detailed analysis of higher regularity obstructions in odd dimensions and proposes a novel method involving a logarithmic variable to achieve concise boundary regularity results.

Findings

Higher regularity is obstructed in odd dimensions.

Introducing the logarithm of the boundary distance as a variable clarifies regularity issues.

The paper establishes concise boundary regularity results for minimal graphs.

Abstract

F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an obstacle to the higher regularity if the dimension is odd. We discuss the odd dimension case and study how the higher regularity is obstructed. We introduce the logarithm of the distance to the boundary as an additional independent self-variable and establish concise boundary regularity for the minimal graph.