  The effects of short-range fermion-fermion interactions on the low-energy
properties of the rhombohedral trilayer graphene are comprehensively
investigated by virtue of the momentum-shell renormalization group method. We
take into account all one-loop corrections and establish the energy-dependent
coupled evolutions of independent fermionic couplings that carry the physical
information stemming from the interplay of various fermion-fermion
interactions. With the help of the detailed numerical analysis, we notice that
the ferocious competition among all fermion-fermion interactions can drive
fermionic couplings to four distinct kinds of fixed points, dubbed
$\textrm{FP}_{1}$, $\textrm{FP}_{2}$, $\textrm{FP}_{3}$ and $\textrm{FP}_{4}$,
in the interaction-parameter space. Such fixed points principally dictate the
fate of the system in the low-energy regime, which are always associated with
some instabilities with specific symmetry breakings and thus accompanied by
certain phase transitions. In order to judge the favorable states arising from
the potential phase transitions, we bring out a number of fermion-bilinear
source terms to characterize the underlying candidate states. By comparing
their related susceptibilities, it is determined that the dominant states
correspond to a spin-singlet superconductivity, a spin-triplet
pair-density-wave, and a spin-triplet superconductivity for approaching the
fixed points $\textrm{FP}_{1,3}$, $\textrm{FP}_{2}$, and $\textrm{FP}_{4}$,
respectively. These results would be helpful to further reveal the low-energy
properties of the rhombohedral trilayer graphene and analogous materials.
