Gravitating vortices with positive curvature

TL;DR

This paper provides a complete existence theory for gravitating vortices with non-negative curvature, solving the Einstein-Bogomol'nyi equations for all admissible Kähler classes and coupling constants, using continuity methods and stability conditions.

Contribution

It establishes the existence of solutions for gravitating vortices with positive and zero topological constants, extending previous results and introducing new bounds on curvature.

Findings

Existence of solutions for c=0 in all admissible Kähler classes.

Existence of solutions for c>0 under GIT stability conditions.

A new curvature bound S_g ≥ c for gravitating vortices.

Abstract

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-Bogomol'nyi equations, corresponding to $c=0$, in all admissible K\"ahler 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 K\"ahler class. For the latter result we start from the established solution in any fixed admissible K\"ahler 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.