New maximal surfaces in minkowski 3-space with arbitrary genus and their cousins in de sitter 3-space

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain "cousins" of those maximal surfaces. Cone-like singular points on maximal surfaces are very important, although they are not stable under perturbations of maximal surfaces. It is interesting to ask if cone-like singular points can appear on a maximal surface having other kinds of singularities. Until now, no such examples were known. We also construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges.