Heisenberg spins on a bilayer connected by a neck and other geometries with a characteristic length scale

We obtain a half-skyrmion solution in the orientation of Heisenberg spins on a neck joining two planes with a semi-circular region. In addition, we consider several geometries, topologically equivalent to either a plane with a hole or a truncated circular cone or a cylinder due to the presence of an intrinsic length scale, for which we obtain skyrmion solutions. We also consider two minimal surfaces, namely a catenoid and a helicoid. Finally, we consider Heisenberg spins on single-sheet paraboloid and hyperboloid geometries. These spin textures may possibly be realized in elastically soft, curved magnetic thin films.