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 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
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 journal › Article
}
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 -