  The film growth of Mn$_3$O$_4$(001) films on Ag(001) up to film thicknesses
of almost seven unit cells of Mn$_3$O$_4$ has been monitored using a
complementary combination of near-edge X-ray absorption fine structure
spectroscopy (NEXAFS), scanning tunneling microscopy (STM), and low-energy
electron diffraction (LEED). The oxide films have been prepared by molecular
beam epitaxy. Using NEXAFS, the identity of the Mn oxide has clearly been
determined as Mn$_3$O$_4$. For the initial stages of growth, oxide islands with
p(2$\times$1) and p(2$\times$2) structures are formed, which are embedded into
the substrate. For Mn$_3$O$_4$ coverages up to 1.5 unit cells a p(2$\times$1)
structure of the films is visible in STM and LEED. Further increase of the
thickness leads to a phase transition of the oxide films resulting in an
additional c(2$\times$2) structure with a 45$^\circ$ rotated atomic pattern.
The emerging film structures are discussed on the basis of a sublayer model of
the Mn$_3$O$_4$ spinel unit cell. While the polarity of the island edges
determines the structure of initial islands, the surface energy of thicker
layers is remarkably reduced by a film restructuring.
