  Motivated by recent experiments on ABC-stacked rhombohedral trilayer graphene
(RTG) which observed spin-valley symmetry-breaking and superconductivity, we
study instabilities of the RTG metallic state to symmetry breaking orders. We
find that interactions select the inter-valley coherent order (IVC) as the
preferred ordering channel over a wide range, whose theoretically determined
phase boundaries agree well with experiments on both the hole and electron
doped sides. The Fermi surfaces near van Hove singularities admit partial
nesting between valleys, which promotes both inter-valley superconductivity and
IVC fluctuations. We investigate the interplay between these fluctuations and
the Hunds (intervalley spin) interaction using a renormalization group
approach. For antiferromagnetic Hund's coupling, intervalley pairing appears in
the spin-singlet channel with enhanced $T_c$, that scales with the
dimensionless coupling $g$ as $T_c\sim\exp(-1/\sqrt{g})$ , compared to the
standard $\exp(-1/g)$ scaling. In its simplest form, this scenario assumes a
sign change in the Hund's coupling on increasing hole doping. On the other
hand, the calculation incorporates breaking of the independent spin rotations
between valleys from the start, and strongly selects spin singlet over spin
triplet pairing, and naturally occurs in proximity to the IVC, consistent with
observations.
