Abstract: The self-dual Einstein-Maxwell-Higgs equations describe a special instance of Einstein's field equations of gravity in four dimensions, coupled to a electromagnetic field and an abelian Higgs field, which saturate a Bogomol'nyi energy bound. Via a natural ansatz, due to Comtet and Gibbons, their solutions can be recast as vortices on a Riemann surface with back-reaction of the metric, known as `gravitating vortices'. In this talk I will overview a novel approach to the existence problem for gravitating vortices based on symplectic reduction by stages. The main technical tool for our study is the reduced ?-K-energy, for which we establish convexity properties by means of finite-energy pluripotential theory. 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 Kähler class, in the absence of automorphisms. Joint work with L. Álvarez-Cónsul, O. García-Prada, V. Pingali, and C. Yao, in arXiv:2406.03639 (to appear in J. Eur. Math. Soc.).