Half-space theorems for \(1\)-surfaces of \(\mathbb{H}^3\) (Q6624031)

An intriguing question in differential geometry is whether two given minimal surfaces in a Riemannian three-manifold intersect. A class of minimal hypersurfaces of a Riemannian manifold is said to have the intersection property if any two elements of the class intersect unless they are totally geodesic parallel leaves in local product structure. For example, the class of complete minimal surfaces with bounded curvature immersed in three-manifolds with positive Ricci curvature and bounded geometry has the intersection property. \N\NThe purpose of this paper is to study the intersection properties of \(1\)-surfaces immersed in hyperbolic space \(\mathbb{H}^{3}\). More generally, the paper studies the intersection properties of \(1\)-surfaces immersed in complete Riemannian three-manifolds with Ricci curvature bounded from below by \(-2\).\N\NTheorem. Let \(P\) be a complete three-manifold with Ricci curvature bounded below by \(-2\). Let \(M\) and \(N\) be complete immersed \(1\)-surfaces of \(P\) that do not intersect and the distance \(\mathrm{dist}(M,N)\) is realized. If \(N\) is proper and \(M\) lies in a mean convex component of \(P \setminus N\), then\N\begin{itemize}\N\item[1.] \(M\) and \(N\) are embedded totally umbilical equidistant \(1\)-surfaces.\N\item[2.] \(M \cup N\) bounds an open connected region in \(P\) whose closure is isometric to \([0, l]\times N\), endowed with the metric \(dt^2+e^{2t}g\), where \(g\) denotes the metric of \(N\).\N\end{itemize}\NIf \(M\) and \(N\) are compact, then each of them is separating, and the mean convex component of \(P\setminus N\) is isometric to \([0, +\infty) \times N\) with metric \(dt^2+e^{2t}g\). In particular, \(P\) cannot be compact.