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In this paper we investigate the intersection problem for $1$-surfacesimmersed in a complete Riemannian three-manifold $P$ with Ricci curvaturebounded from below by $-2$. We first prove a Frankel's type theorem for$1$-surfaces with bounded curvature immersed in $P$ when $\text{\rm Ric}_{P} >-2$. In this setting we also give a criterion for deciding whether a complete$1$-surface is proper. A splitting result is established when the distancebetween the $1$-surfaces is realized, even if $\text{\rm Ric}_{P} \geq -2$. Inthe hyperbolic space $\mathbb{H}^3$ we show strong half-space theorems for theclasses of complete $1$-surfaces with bounded curvature, parabolic$1$-surfaces, and stochastically complete $H$-surfaces with $H<1$. As aby-product of our techniques a Maximum Principle at Infinity is given for$1$-surfaces in $\mathbb{H}^3.$

Half-space theorems for $1$-surfaces of $\mathbb{H}^3$