Node-Opening Construction of Maxfaces

The node-opening technique developed by Traizet has been very useful in constructing minimal surfaces. In this paper, we use the technique to construct families of maximal immersions in Lorentz space that are embedded outside a compact set. Each family depends on a real parameter t. The surfaces look like horizontal planes connected by small necks that shrink to singular points as t 0. The limit positions of the necks must satisfy a balance condition, which turns out to be exactly the same for maxfaces and for minimal surfaces. By simply comparing notes, we obtain a rich variety of new maxfaces with high genus and arbitrarily many space-like ends. Among them are the Lorentzian Costa--Hofmann--Meeks surfaces. Non-planar complete maximal immersions must have singularities. We will analyse the singularity structure. For sufficiently small non-zero t, the singular set consists of curves in the waist of every neck. In generic and some symmetric cases, we are able to prove that all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails.