On the asymptotic Plateau problem in SL˜2(R)

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 + 4 τ 2 π 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 ) .