  In this paper, we consider the self-dual equation arising from Abelian
Chern-Simons-Higgs theory coupled to the Einstein equations over the plane
$\mathbb{R}^2$ 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
$\chi(S)$ of $S$, and the gravitational coupling factor $G$ is given to show
the existence of $N$-string solutions over a compact surface.
