On the asymptotic behavior of minimal surfaces in H${}^2$$\times$R

TL;DR

This paper studies the asymptotic properties of properly embedded minimal surfaces in hyperbolic plane times a line, exploring different compactifications, constructing new examples, and analyzing boundary regularity.

Contribution

It introduces a refined framework for understanding minimal surfaces in H^2×R, including new examples and boundary regularity results.

Findings

Different compactifications affect the asymptotic boundary analysis.

Constructed new minimal surface examples in H^2×R.

Described boundary regularity conditions for these surfaces.

Abstract

We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one natural compactification of this space. This provides a better setting in which to consider the general problem of determining which curves at infinity are the asymptotic boundary of such minimal surfaces. We also construct some new examples of such surfaces and describe the boundary regularity.