Geodesic Boundary of Parabolic Surfaces and Existence of H-Fillable Curves in H2×R

This article provides a geometric characterization of the geodesic boundary for surfaces invariant under parabolic isometries in H2×R. We present an alternative, constructive proof for the existence of minimal surfaces with rectangular asymptotic boundaries by utilizing a specific family of invariant surfaces. Furthermore, we generalize these existence results to surfaces with constant mean curvature H∈(0,1/2). By analyzing the variation in the relative asymptotic height, we establish the existence of properly embedded H-surfaces whose geodesic boundary is a rectangle of arbitrary height, provided it exceeds a specific lower bound determined by the parabolic solution.