Electronic structure and magnetism of the triple-layered ruthenate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Sr</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Ru</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>10</mml:mn></mml:msub></mml:math>
We report electronic band structure calculations for ${\mathrm{Sr}}_{4}{\mathrm{Ru}}_{3}{\mathrm{O}}_{10}$ which display both ferromagnetic and metamagnetic behavior. The density functional calculations find the ground state to be ferromagnetic in agreement with experiment and we show that the resulting spin polarization has dramatic consequences for the electronic properties. The minority-spin bands are mainly empty and disperse steeply upward at the Fermi energy, whereas the majority-spin bands are full or nearly fully occupied and form narrow bands near the Fermi energy, which could be the electronic origin of the metamagnetism. Inclusion of a Hubbard interaction $U$ applied to the Ru $4d$ states has major effects on the narrow bands, which reveal the role of Coulomb interactions and correlated many-body physics. 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.