Obstructions to the existence of solutions of the self-dual Einstein-Maxwell-Higgs equations on a compact surface

TL;DR

This paper identifies conditions based on zero multiplicities and vortex number that prevent solutions to the self-dual Einstein-Maxwell-Higgs equations on compact surfaces, providing numerous examples where solutions do not exist.

Contribution

It introduces an obstruction criterion depending on zero multiplicities and vortex number, showing that solutions cannot exist for infinitely many Higgs fields.

Findings

Obstruction depends on zero multiplicities and vortex number

Infinitely many Higgs fields lack solutions due to this obstruction

Provides explicit examples of non-existence cases

Abstract

In this note we present an obstruction to the existence of solutions to the self-dual Einstein-Maxwell-Higgs equations on a compact surface, which depends on the multiplicities of the zeroes of the \emph{Higgs field} and the \emph{vortex number} $N$. In particular, we exhibit infinitely many examples of Higgs fields for which solutions cannot exist.