We study the ferroelectric stability and surface structural properties of an oxygen-terminated hexagonal YMnO3 ultra-thin film using density functional theory. Under an open circuit boundary condition, the ferroelectric state with the spontaneous polarization normal to the (0001) surface, is found to be metastable in a single domain state despite the presence of a depolarizing field. We establish a connection between the result and the role of improper ferroelectric transition. Our results imply that improper ferroelectric ultrathin films can have rather unique properties that are distinctive from those of very thin films of ordinary ferroelectrics.The integration of ferroelectric oxide materials into existing microelectronic device architectures is currently of much interest as they hold promise for a wide range of potentially new applications [1,2]. This has generated an enormous effort to understand the properties of ferroelectric perovskite ultrathin films, such as BaTiO 3 , where it has been shown that electrical boundary conditions [3,4,5,6], surface and interface properties [7,8,9,10], as well as epitaxial strain [11,12] all play an important role. Although there appears to be no fundamental size limit below which ferroelectricity disappears, ultrathin ferroelectric films with the polarization normal to the surface remain a challenge. This is because the depolarization field arising from the accumulated charges at the surfaces, if not screened, can strongly suppress the instability towards a single-domain ferroelectric state. This is indeed what happens in a material displaying a proper ferroelectric transition (note, henceforth we refer to such materials as proper ferroelectrics), namely one where the primary order parameter is the electrical polarization, e.g., BaTiO 3 . The depolarizing field contributes to the free energy a positive term quadric in the polarization thereby re-normalizing the soft-mode energy and stabilizing the paraelectric phase. Even in a system with metallic electrodes, which can provide the necessary screening in most cases, when the thickness of the ferroelectric film becomes comparable to the effective screening length of the metal, the screening is incomplete resulting in a reduced polarization [5,7,13], an increased coercive field [14], and in some case suppress the tendency towards (single-domain) ferroelectricity completely.It would clearly be advantageous, if not at least fundamentally interesting, to consider the surface properties of ultrathin films of materials in which ferroelectricity did not originate from a polar instability, but rather from an improper ferroelectric transition -i.e., one where the spontaneous polarization does not drive the transition but instead is a slave to some other primary order parameter. As pointed out decades ago by Levanyuk and San- nikov in a series of papers [15,16], an instability towards a single-domain ferroelectric state in a improper ferroelectric is still possible even if the depolarization field remains unscreened, e.g., under ope...