Gravitating vortices with positive curvature

Abstract

We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c⩾0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomol'nyi equations, corresponding to $c=0$, in all admissible Kähler 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 Kähler class. For the latter result we start from the established solution in any fixed admissible Kähler class and deform the coupling constant $\alpha$ towards $0$. A salient feature of our argument is a new bound $S_g⩾c$ for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

31 pages. New Theorem 1.1, where we prove the existence of solutions to the Einstein-Bogomol'nyi equations/self-dual Einstein-Maxwell-Higgs equations in all admissible Kähler classes. Introduction and abstract modified. References updated