Multiple Delaunay ends solutions of the Cahn-Hilliard equation

Let $\mathrm{\Sigma }$ be a surface of constant mean curvature in ${\mathbb{R}}^3$ with multiple Delaunay ends. Assuming that $\mathrm{\Sigma }$ is non degenerate in this paper we construct new solutions to the Cahn-Hilliard equation $\varepsilon\Delta u+\varepsilon^{-1}u(1-u^2)=\ell_\varepsilon$in${\mathbb R}^3$suchthatas$\varepsilon\to 0$thezerolevelsetof$u_\varepsilon$approaches$\Sigma$. Moreover, on compacts of the connected components of ${\mathbb R}^3\setminus \Sigma$wehave$1-|u_\varepsilon|\to 0$ uniformly.