### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 41-82 |

Number of pages | 42 |

Journal | Results in Mathematics |

Volume | 56 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Jan 1 |

### Fingerprint

### Keywords

- CMC-1 surfaces in de sitter space
- Maximal surfaces
- Singularities

### ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Applied Mathematics

### Cite this

*Results in Mathematics*,

*56*(1), 41-82. https://doi.org/10.1007/s00025-009-0443-4

**New maximal surfaces in minkowski 3-space with arbitrary genus and their cousins in de sitter 3-space.** / Fujimori, Shoichi; Rossman, Wayne; Umehara, Masaaki; Yamada, Kotaro; Yang, Seong-Deog.

Research output: Contribution to journal › Article

*Results in Mathematics*, vol. 56, no. 1, pp. 41-82. https://doi.org/10.1007/s00025-009-0443-4

}

TY - JOUR

T1 - New maximal surfaces in minkowski 3-space with arbitrary genus and their cousins in de sitter 3-space

AU - Fujimori, Shoichi

AU - Rossman, Wayne

AU - Umehara, Masaaki

AU - Yamada, Kotaro

AU - Yang, Seong-Deog

PY - 2009/1/1

Y1 - 2009/1/1

N2 - 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.

AB - 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.

KW - CMC-1 surfaces in de sitter space

KW - Maximal surfaces

KW - Singularities

UR - http://www.scopus.com/inward/record.url?scp=72149115938&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72149115938&partnerID=8YFLogxK

U2 - 10.1007/s00025-009-0443-4

DO - 10.1007/s00025-009-0443-4

M3 - Article

VL - 56

SP - 41

EP - 82

JO - Results in Mathematics

JF - Results in Mathematics

SN - 1422-6383

IS - 1

ER -