# Properly embedded minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$

## TL;DR

This paper investigates the moduli space of properly embedded minimal annuli in hyperbolic space cross the real line, revealing limitations on boundary data prescription and establishing existence results with symmetry constraints.

## Contribution

It demonstrates that boundary curves at infinity cannot be fully prescribed and provides existence theorems for symmetric minimal annuli in   imes .

## Key findings

- Boundary curves at infinity are only partially prescribable.
- Existence of minimal annuli with specified symmetries is established.
- The top boundary curve is determined up to translation and tilt.

## Abstract

In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$ with horizontal ends. We say that the ends are horizontal when they are graphs of $\mathcal{C}^{2, \alpha}$ functions over $\partial_\infty \mathbb{H}^2$. Contrary to expectation, we show that one can not fully prescribe the two boundary curves at infinity, but rather, one can prescribe the bottom curve, but the top curve only up to a translation and a tilt, along with the position of the neck and the vertical flux of the annulus. We also prove general existence theorems for minimal annuli with discrete groups of symmetries.