We show the existence of a complete, strictly locally convex hypersurface within \({\mathbb {H}}\) that adheres to a curvature equation applicable to a broad range of curvature functions. This hypersurface possesses a prescribed asymptotic boundary at infinity and takes the form of a geodesic graph over a smooth bounded domain \(\Omega \) at infinity. It is approximated by the shape of geodesic graphs whose boundaries rest upon equidistant hyperplanes. Through this procedure, we establish an alternative method for constructing solutions to the asymptotic Plateau problem comparing to Guan–Spruck–Szapiel–Xiao (J Geom Anal 19(4):772–795, 2009. https://doi.org/10.1007/s12220-009-9086-7; J Differ Geom 96(2):201–222, 2014).
