New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space. (English) Zbl 1186.53071
Maximal surfaces in the Minkowski 3-space \(\mathbb{R}_{1}^{3}\) arise as solutions of the variational problem of locally maximizing the area among space-like surfaces. Until now, the only known maximal surfaces in \(\mathbb{R}_{1}^{3}\) 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 the present paper, the authors discuss singularities and global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, the maximal surfaces in the examples are embedded outside a compact set of \(\mathbb{R}_{1}^{3}.\) Moreover, these maximal surfaces can be deformed to \(CMC-1\) surfaces (mean curvature one surfaces with admissible singularities in de Sitter 3-space). All these results are stated in the Theorem B of the paper.
In the Theorem A of the paper the authors construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges. Notice that until now, no such examples were known.
In the Theorem A of the paper the authors construct a family of complete maximal surfaces with two complete ends and with both cone-like singular points and cuspidal edges. Notice that until now, no such examples were known.
Reviewer: Charalampos Charitos (Athens)
MSC:
53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
53A35 | Non-Euclidean differential geometry |
53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |