  Bulk BiMnO$_3$ is the only transition-metal perovskite oxide that is
insulating and shows strong ferromagnetism. This distinctive behavior would
make it a promising candidate as a magnetoelectric multiferroic if it was also
a polar material, but experiments have shown that bulk BiMnO$_3$ has either a
very small polarization (below 0.1~$\mu$C/cm$^2$) or, most likely, that it is a
paraelectric. There is also experimental evidence that the polarization in
BiMnO$_3$ {\em films} grown on SrTiO$_3$ can be as high as 20~$\mu$C/cm$^2$.
Despite of the interest of these behaviors, the diagram of BiMnO$_3$ as a
function of epitaxial strain has remained largely unexplored. In this article,
we use first-principles to predict that both under enough compressive and
tensile epitaxial strain BiMnO$_3$ films are ferroelectric with a giant
polarization around 100~$\mu$C/cm$^2$. The phases displayed by the films are
similar to those experimentally found for BiFeO$_3$ in similar conditions---at
compressive strains, the film is supertetragonal with a large component of the
polarization pointing out of plane, while at tensile strains the polarization
points mostly in plane. Like in BiFeO$_3$ films, these phases are
antiferromagnetic---the orbital ordering responsible for ferromagnetism in
BiMnO$_3$ is absent in the polar phases. Our calculations also show that the
band gap of some of these BiMnO$_3$ films is substantially smaller than gaps
typically found in ferroelectric oxides, suggesting it may be a suitable
material for photovoltaic applications.
