We demonstrate the fabrication of thin GaAs layers which quasi hover above the underlying GaAs substrate. The hovering layers have a perfect epitaxial relationship to the substrate crystal lattice and are connected to the substrate surface only by lattice matched nanopillars of low density. These air-gap heterostructures are created by combining in situ molecular beam epitaxy compatible self-assembled droplet-etching and ex situ selective wet-chemical etching.