Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RULED MINIMAL SURFACES IN PRODUCT SPACES

  • Jin, Yuzi (Department of Mathematics Jilin Institute of Chemical Technology) ;
  • Kim, Young Wook (Department of Mathematics Korea University) ;
  • Park, Namkyoung (Department of Mathematics Chung-Ang University) ;
  • Shin, Heayong (Department of Mathematics Chung-Ang University)
  • Received : 2016.01.04
  • Published : 2016.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

It is well known that the helicoids are the only ruled minimal surfaces in ${\mathbb{R}}^3$. 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 $M{\times}{\mathbb{R}}$ for a 2-dimensional manifold M and prove that $M{\times}{\mathbb{R}}$ 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 ${\mathbb{R}}$.

Keywords

References

  1. M. Do Carmo and M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 (1983), no. 2, 685-709. https://doi.org/10.1090/S0002-9947-1983-0694383-X
  2. Y. W. Kim, S.-E. Koh, H. Shin, and S.-D. Yang, Helicoids in ${\mathbb{S}}^2{\times}{\mathbb{R}}\;and\;{\mathbb{H}}^2{\times}{\mathbb{R}}$, Pacific J. Math. 242 (2009), no. 2, 281-297. https://doi.org/10.2140/pjm.2009.242.281
  3. H. B. Lawson, Complete minimal surfaces in $S^3$, Ann. Math. 92 (1970), 335-374. https://doi.org/10.2307/1970625
  4. B. O'neill, Elementary Differential Geometry, 2nd Ed., Academic Press, 1997.
  5. H. Shin, Y. W. Kim, S.-E. Koh, H. Y. Lee, and S.-D. Yang, Ruled minimal surfaces in the three dimensional Heisenberg group, Pacific J. Math. 261 (2013), no. 2, 477-496. https://doi.org/10.2140/pjm.2013.261.477
  6. H. Shin, Y. W. Kim, S.-E. Koh, H. Y. Lee, and S.-D. Yang, Ruled minimal surfaces in the Berger sphere, Differential Geom. Appl. 40 (2015), 209-222. https://doi.org/10.1016/j.difgeo.2015.02.007
  7. S.-D. Yang, Y. W. Kim, S.-E. Koh, H. Y. Lee, and H. Shin, Ruled minimal surfaces in $SL_2{\mathbb{R}}$, Preprint.

Cited by

  1. Minimal rotational surfaces in the product space ℚ𝜀2 × 𝕊1 vol.29, pp.08, 2018, https://doi.org/10.1142/S0129167X18500519