Ruled minimal surfaces in product spaces

Yuzi Jin, Young Wook Kim, Namkyoung Park, Heayong Shin

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1887-1892
Number of pages6
JournalBulletin of the Korean Mathematical Society
Volume53
Issue number6
DOIs
Publication statusPublished - 2016

Fingerprint

Ruled Surface
Product Space
Minimal surface
Longitude
Homogeneous Space
Parametrization
Geodesic
Rotating
Horizontal
Trace

Keywords

  • Helicoid
  • Minimal surface
  • Ruled surface

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Bulletin of the Korean Mathematical Society, Vol. 53, No. 6, 2016, p. 1887-1892.

Research output: Contribution to journalArticle

Jin, Yuzi ; Kim, Young Wook ; Park, Namkyoung ; Shin, Heayong. / Ruled minimal surfaces in product spaces. In: Bulletin of the Korean Mathematical Society. 2016 ; Vol. 53, No. 6. pp. 1887-1892.
@article{1fde39857abd4c4c9ab427ded9c2898c,
title = "Ruled minimal surfaces in product spaces",
abstract = "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.",
keywords = "Helicoid, Minimal surface, Ruled surface",
author = "Yuzi Jin and Kim, {Young Wook} and Namkyoung Park and Heayong Shin",
year = "2016",
doi = "10.4134/BKMS.b160006",
language = "English",
volume = "53",
pages = "1887--1892",
journal = "Bulletin of the Korean Mathematical Society",
issn = "1015-8634",
publisher = "Korean Mathematical Society",
number = "6",

}

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 -