  We undertake a novel approach to the existence problem for gravitating
vortices on a Riemann surface based on symplectic reduction by stages, which
seems to be new in the PDE as well as the gauge theory literature. The main
technical tool for our study is the reduced $\alpha$-K-energy, for which we
establish convexity properties by means of finite-energy pluripotential theory,
as recently applied to the study of constant scalar curvature K\"ahler metrics.
Using these methods, we prove that the existence of solutions to the
gravitating vortex equations on the sphere implies the polystability of the
effective divisor defined by the zeroes of the Higgs field. This approach also
enables us to establish the uniqueness of gravitating vortices in any
admissible K\"ahler class, in the absence of automorphisms. Lastly, we also
prove the existence of solutions for the gravitating vortex equations for genus
$g\geq 1$ for certain ranges of the coupling constant $\alpha$ and the volume.
