Multiple cosmic strings in Chern–Simons–Higgs theory with gravity

In this paper, we consider the self-dual equation arising from Abelian Chern–Simons–Higgs theory coupled to the Einstein equations over the plane $R^2$ and a compact surface $S$. We prove the existence of symmetric topological solutions and non-topological solutions on the plane by using the fixed-point theorem and a shooting method, respectively. A necessary and sufficient condition related to the string number $N$, the Euler characteristic $\chi \left(S\right)$ of $S$, and the gravitational coupling factor $G$ is given to show the existence of $N$-string solutions over a compact surface.