### Abstract

It is well known that the helicoids are the only ruled minimal surfaces in ℝ^{3}. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space M × ℝ for a 2-dimensional manifold M and prove that M ×ℝ has a nontrivial minimal surface ruled by horizontal geodesics only when M has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in M while translating vertically in constant speed in the direction of ℝ.

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

Pages (from-to) | 1887-1892 |

Number of pages | 6 |

Journal | Bulletin of the Korean Mathematical Society |

Volume | 53 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2016 |

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### Keywords

- Helicoid
- Minimal surface
- Ruled surface

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Korean Mathematical Society*,

*53*(6), 1887-1892. https://doi.org/10.4134/BKMS.b160006

**Ruled minimal surfaces in product spaces.** / Jin, Yuzi; Kim, Young Wook; Park, Namkyoung; Shin, Heayong.

Research output: Contribution to journal › Article

*Bulletin of the Korean Mathematical Society*, vol. 53, no. 6, pp. 1887-1892. https://doi.org/10.4134/BKMS.b160006

}

TY - JOUR

T1 - Ruled minimal surfaces in product spaces

AU - Jin, Yuzi

AU - Kim, Young Wook

AU - Park, Namkyoung

AU - Shin, Heayong

PY - 2016

Y1 - 2016

N2 - It is well known that the helicoids are the only ruled minimal surfaces in ℝ3. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space M × ℝ for a 2-dimensional manifold M and prove that M ×ℝ has a nontrivial minimal surface ruled by horizontal geodesics only when M has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in M while translating vertically in constant speed in the direction of ℝ.

AB - It is well known that the helicoids are the only ruled minimal surfaces in ℝ3. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space M × ℝ for a 2-dimensional manifold M and prove that M ×ℝ has a nontrivial minimal surface ruled by horizontal geodesics only when M has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in M while translating vertically in constant speed in the direction of ℝ.

KW - Helicoid

KW - Minimal surface

KW - Ruled surface

UR - http://www.scopus.com/inward/record.url?scp=84996569809&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84996569809&partnerID=8YFLogxK

U2 - 10.4134/BKMS.b160006

DO - 10.4134/BKMS.b160006

M3 - Article

AN - SCOPUS:84996569809

VL - 53

SP - 1887

EP - 1892

JO - Bulletin of the Korean Mathematical Society

JF - Bulletin of the Korean Mathematical Society

SN - 1015-8634

IS - 6

ER -