Minimal Graphs in the Hyperbolic Space with Singular Asymptotic Boundaries

TL;DR

This paper investigates the asymptotic behavior of minimal graphs in hyperbolic space with singular boundaries, providing estimates based on tangent ball intersections, emphasizing the importance of boundary curvature positivity.

Contribution

It introduces new estimates for minimal graphs with singular asymptotic boundaries in hyperbolic space, highlighting the role of boundary curvature positivity.

Findings

Derived estimates for solutions using tangent ball intersections

Established the significance of positive boundary curvature

Analyzed solutions with singular asymptotic boundaries

Abstract

We study asymptotic behaviors of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures. We derive an estimate of such solutions by the corresponding solutions in the intersections of interior tangent balls. The positivity of curvatures plays an important role.