Half-Space Theorems for \(\textbf{1}\)-Surfaces of \(\mathbb {H}^3\)

We study strong half-space theorems for the classes of complete 1-surfaces with bounded curvature, parabolic 1-surfaces, and stochastically complete H-surfaces with \(H<1\) $H<1$ immersed in the hyperbolic space \(\mathbb {H}^3\) ${\mathbb{H}}^3$ . As a by-product of the techniques we obtain a Maximum Principle at Infinity for 1-surfaces in \(\mathbb {H}^3\) ${\mathbb{H}}^3$ . We also address the intersection problem for 1-surfaces immersed in a complete Riemannian three-manifold P with Ricci curvature bounded from below by \(-2\) $-2$ . We establish a splitting result provided the distance between the 1-surfaces is realized and \(\text {Ric}_{P} \ge -2\) ${\text{Ric}}_P\ge -2$ , and a Frankel’s type theorem for 1-surfaces with bounded curvature immersed in P when \(\text { Ric}_{P} > -2\) ${\text{Ric}}_P>-2$ .