  We study the ground state properties of an ABA-stacked trilayer graphene. The
low energy band structure can be described by a combination of both a linear
and a quadratic particle-hole symmetric dispersions, reminiscent of monolayer-
and bilayer-graphene, respectively. The multi-band structure offers more
channels for instability towards ferromagnetism when the Coulomb interaction is
taken into account. Indeed, if one associates a pseudo-spin 1/2 degree of
freedom to the bands (parabolic/linear), it is possible to realize also a
band-ferromagnetic state, where there is a shift in the energy bands, since
they fill up differently. By using a variational procedure, we compute the
exchange energies for all possible variational ground states and identify the
parameter space for the occurrence of spin- and band-ferromagnetic
instabilities as a function of doping and interaction strength.
