Singularities on maxfaces constructed by node-opening

Abstract: The node-opening technique, originally designed for constructing minimal surfaces, is adapted to construct a rich variety of new maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar ends, thus removing the scarcity of examples of maxfaces. The surfaces look like spacelike planes connected by small necks. Among the examples are maxfaces of the Costa--Hoffman--Meeks type. Although very fruitful, the main challenge of this paper is not the construction itself, but the analysis of the positions and natures of singularities on these maxfaces. More specifically, we conclude that the singular set form curves around the waists of the necks. In generic and some symmetric cases, all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails evenly distributed along the singular curves.