DOI QR코드

DOI QR Code

HELICOIDAL KILLING FIELDS, HELICOIDS AND RULED MINIMAL SURFACES IN HOMOGENEOUS THREE-MANIFOLDS

  • 투고 : 2017.10.20
  • 심사 : 2018.04.09
  • 발행 : 2018.09.01

초록

We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $SL_2{\mathbb{R}}$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.

키워드

참고문헌

  1. M. Bekkar, F. Bouziani, Y. Boukhatem, and J. Inoguchi, Helicoids and axially symmetric minimal surfaces in 3-dimensional homogeneous spaces, Differ. Geom. Dyn. Syst. 9 (2007), 21-39.
  2. S. Cartier, Noether invariants for constant mean curvature surfaces in 3-dimensional homogeneous spaces, arXiv:1303.6391v1.
  3. B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), no. 1, 87-131.
  4. S. Fujimori, W. Rossman, M. Umehara, K. Yamada, and S.-D. Yang, Spacelike mean curvature one surfaces in de Sitter 3-space, Comm. Anal. Geom. 17 (2009), no. 3, 383-427. https://doi.org/10.4310/CAG.2009.v17.n3.a1
  5. 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
  6. Y. W. Kim, S.-E. Koh, H. Shin, and S.-D. Yang, Helicoidal minimal surfaces in ${\mathbb{H}}^2{\times}{\mathbb{R}}$, Bull. Aust. Math. Soc. 86 (2012), no. 1, 135-149. https://doi.org/10.1017/S0004972711003042
  7. M. Kokubu, On minimal surfaces in the real special linear group SL(2,R), Tokyo J. Math. 20 (1997), no. 2, 287-297. https://doi.org/10.3836/tjm/1270042104
  8. H. B. Lawson, Jr., Complete minimal surfaces in $S^3$, Ann. of Math. (2) 92 (1970), 335-374. https://doi.org/10.2307/1970625
  9. R. Lopez and A. I. Nistor, Surfaces in $Sol_3$ space foliated by circles, Results Math. 64 (2013), no. 3-4, 319-330. https://doi.org/10.1007/s00025-013-0316-8
  10. J. Plehnert, Constant mean curvature k-noids in homogeneous manifolds, Illinois J. Math. 58 (2014), no. 1, 233-249.
  11. 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
  12. 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
  13. R. Souam and E. Toubiana, Totally umbilic surfaces in homogeneous 3-manifolds, Comment. Math. Helv. 84 (2009), no. 3, 673-704.