New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space

摘要

Until now, the only known maximal surfaces in Minkowski 3-space of finitetopology with compact singular set and without branch points were either genuszero or genus one, or came from a correspondence with minimal surfaces inEuclidean 3-space given by the third and fourth authors in a previous paper. Inthis paper, we discuss singularities and several global properties of maximalsurfaces, and give explicit examples of such surfaces of arbitrary genus. Whenthe genus is one, our examples are embedded outside a compact set. Moreover, wedeform such examples to CMC-1 faces (mean curvature one surfaces withadmissible singularities in de Sitter 3-space) and obtain "cousins" of thosemaximal surfaces. Cone-like singular points on maximal surfaces are very important, althoughthey are not stable under perturbations of maximal surfaces. It is interestingto ask if cone-like singular points can appear on a maximal surface havingother kinds of singularities. Until now, no such examples were known. We alsoconstruct a family of complete maximal surfaces with two complete ends and withboth cone-like singular points and cuspidal edges.