  The intrinsic magnetic state (ferromagnetic or antiferromagnetic) of
ultra-thin LaMnO$_3$ films on the mostly used SrTiO$_3$ substrate is a
long-existing question under debate. Either strain effect or non-stoichiometry
was argued to be responsible for the experimental ferromagnetism. In a recent
experiment [Science \textbf{349}, 716 (2015)], one more mechanism, namely the
self-doping due to polar discontinuity, was argued to be the driving force of
ferromagnetism beyond the critical thickness. Here systematic first-principles
calculations have been performed to check these mechanisms in ultra-thin
LaMnO$_3$ films as well as superlattices. Starting from the very precise
descriptions of both LaMnO$_3$ and SrTiO$_3$, it is found that the compressive
strain is the dominant force for the appearance of ferromagnetism, while the
open surface with oxygen vacancies leads to the suppression of ferromagnetism.
Within LaMnO$_3$ layers, the charge reconstructions involve many competitive
factors and certainly go beyond the intuitive polar catastrophe model
established for LaAlO$_3$/SrTiO$_3$ heterostructures. Our study not only
explains the long-term puzzle regarding the magnetism of ultra-thin LaMnO$_3$
films, but also shed light on how to overcome the notorious magnetic dead layer
in ultra-thin manganites.
