Some structure theorems for complete \(H\)-surfaces in hyperbolic 3-space \(\mathbb{H}^3\)
We consider properly embedded complete noncompact (nonzero) constant mean curvature surfaces \(M\) of finite topology in hyperbolic 3-space with boundary a strictly convex curve \(C\) in a geodesic plane \(P\). Assume \(M\) is transverse to \(P\) along \(C\) and has a finite number of cylindrically bounded ends orthogonal to \(P\) contained in one of the half-spaces determined by \(P\). We prove that if \(M\) is contained in the half-space, then \(M\) inherits the symmetries of \(C\). In particular, \(M\) is a Delaunay surface if \(C\) is a circle. Then we give conditions that ensure that if \(M\) is contained in the half-space only near \(C\), \(M\) is entirely in the half-space.
noncompact surfaces of constant mean curvature with boundary
scientific article; zbMATH DE number 1191159