Minimal Surfaces in $\mathbb{H}_2\times \mathbb{R}$: Nonfillable Curves

TL;DR

This paper investigates the asymptotic Plateau problem in hyperbolic space times a line, providing new examples of non-fillable curves and analyzing fillability conditions for infinite curves at infinity.

Contribution

It presents the first examples of non-fillable finite curves without thin tails and explores fillability criteria for infinite curves in the asymptotic boundary.

Findings

Constructed non-fillable finite curves with no thin tail.

Analyzed fillability conditions for infinite curves.

Extended understanding of boundary behavior in hyperbolic product spaces.

Abstract

We study the asymptotic Plateau problem in $\mathbb{H}_2\times \mathbb{R}$. We give the first examples of non-fillable finite curves with no thin tail in the asymptotic cylinder. Furthermore, we study the fillability question for infinite curves in the asymptotic boundary.