We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c geqslant 0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomolnyi equations, corresponding to $c=0$, in all admissible Kahler classes. Our second main result completely solves the existence problem for $c>0$. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with $c = 0$ and deforms the Kahler class. For the latter result we start from the established solution in any fixed admissible Kahler class and deform the coupling constant $alpha$ towards $0$. A salient feature of our argument is a new bound $S_g geqslant c$ for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.
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