On the self-dual Einstein-Maxwell-Higgs equations from the Abelian-Higgs model

In this talk, we consider the self-dual equations arising from the Einstein-Maxwell-Higgs model that is regarded as a general reletivistic version of the classical Ginzburg-Landau theory.
We introduce the Lagrangian density and explain how we reduce it to two dimensional problem.
Especially, we derive the self-dual equations coupled with the Einstein equations.
Then, we review  some known results about the existence of solutions on $\mathbb{R}^2$ and on compact surfaces with brief ideas of proof.