We investigate the occurrence of magnetic and charge density wave instabilities in rhombohedral stacked multilayer (3 to 8 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 $10^{-3}μ_B$), 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 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 $n ∼ 7 \times 10^{11}$ cm$^{-2}$ the gap closes. Charge density wave instabilities with $\sqrt{3}\times\sqrt{3}$ periodicity do not occur. ; Comment: 5 pages, 5 figures