Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In particular I am interested in the case of a infinite cylinder $S^1 \times \mathbb{R}$.

For finiteenergy vortices on a finitetype Riemannian surface with cylindrical ends, there is still a nonnegative integer parameter, the vortex number $N$, and the moduli space is still canonically diffeomorphic to the $N$th symmetric product by the map that takes a gaugeequivalence class of vortices $[A,\phi]$ to $\phi^{1}(0)$. One can prove that such vortices extend over the puncture, whereupon the usual methods apply. Some references: 1) The case of the complex plane was treated in the book "Vortices and monopoles" by JaffeTaubes. 2) The case of a cylinder is explicitly treated, by a different method, in a paper by Frauenfelder: http://arxiv.org/abs/math/0507285 3) One can regard the vortex equations as dimensional reductions of the SeibergWitten equations. There is a comprehensive treatment of those equations in the presence of cylindrical ends in Kronheimer and Mrowka's book "Monopoles and 3manifolds". They also discuss AtiyahPatodiSinger boundary conditions in the case where there is a boundary. 

