# On the asymptotic Plateau problem for area minimizing surfaces in $\mathbb{E}(-1,\tau)$

## TL;DR

This paper investigates the existence of complete area minimizing surfaces in the homogeneous space (-1, au), providing conditions under which certain boundary curves guarantee solutions to the asymptotic Plateau problem.

## Contribution

It establishes new sufficient conditions for the existence of area minimizing surfaces with prescribed asymptotic boundaries in (-1, au).

## Key findings

- Identifies conditions for existence of solutions to the asymptotic Plateau problem.
- Provides non-existence results under certain boundary configurations.
- Advances understanding of minimal surfaces in homogeneous spaces.

## Abstract

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,\tau)$. As one of our main results, we present sufficient conditions for a curve $\Gamma$ in $\partial_{\infty} \mathbb{E}(-1,\tau)$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,\tau)$ having $\Gamma$ as its asymptotic boundary.