  We investigate the occurrence of magnetic and charge density wave
instabilities in rhombohedral-stacked multilayer (three to eight layers)
graphene by first principles calculations including exact exchange. Neglecting
spin-polarization, an extremely flat surface band centered at the special point
${\bf K}$ of the Brillouin zone occurs at the Fermi level. Spin polarization
opens a gap in the surface state by stabilizing an antiferromagnetic state. The
top and the bottom surface layers are weakly ferrimagnetic in-plane (net
magnetization smaller than $10^{-3}\mu_B$), and are antiferromagnetic coupled
to each other. This coupling is propagated by the out-of-plane
antiferromagnetic coupling between the nearest neighbors. The gap is very small
in a spin-polarized generalized gradient approximation, while it is
proportional to the amount of exact exchange in hybrid functionals. For
trilayer rhombohedral graphene it is $38.6$ meV in PBE0, in agreement with the
$42$ meV gap found in experiments. We study the temperature and doping
dependence of the magnetic gap. At electron doping of $n \sim 7 \times 10^{11}$
cm$^{-2}$ the gap closes. Charge density wave instabilities with
$\sqrt{3}\times\sqrt{3}$ periodicity do not occur.
