  We report electronic band structure calculations for Sr$_{4}$Ru$_{3}$O$_{10}$
that displays both ferromagnetic and metamagnetic behavior. The density
functional calculations find the ground state to be ferromagnetic in agreement
with the experiment and we find that the inclusion of Coulomb Hubbard
interaction U applied to the Ru 4d states has dramatic effects on the Fermi
surface, which reveal the role of Coulomb interactions and correlated many-body
physics. The minority spin bands are mainly empty with Fermi surfaces in the
outer areas of the Brillouin zone away from the $\Gamma$ point with bands that
disperse steeply upward. The majority spin bands are full or nearly fully
occupied and form narrow bands near the Fermi energy around the $\Gamma$ point,
which could be the electronic origin of the metamagnetism. The results are in
qualitative agreement with recent angle resolved photoemission spectroscopy
(ARPES) experiments and show the need for a combined theoretical study and
experimental ARPES investigation with better energy resolution to reveal the
nature of the narrow bands close to the Fermi-level, which is critical for
understanding the exotic magnetic properties observed in this material.
