On the asymptotic Plateau problem in [formula omitted

Abstract

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space SL˜2(R) with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in SL˜2(R) contained between two bounded entire minimal graphs separated by vertical distance less than 1+4τ2π has multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in SL˜2(R).