Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE GAUSS MAP OF HELICOIDAL SURFACES

  • Kim, Dong-Soo (Department of Mathematics Chonnam National University) ;
  • Kim, Wonyong (Department of Mathematics Chonnam National University) ;
  • Kim, Young Ho (Department of Mathematics Kyungpook National University)
  • Received : 2016.09.13
  • Accepted : 2016.12.22
  • Published : 2017.07.31
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Abstract

We study the Gauss map G of helicoidal surfaces in the 3-dimensional Euclidean space ${\mathbb{E}}^3$ with respect to the so called Cheng-Yau operator ${\square}$ acting on the functions defined on the surfaces. As a result, we completely classify the helicoidal surfaces with Gauss map G satisfying ${\square}G=AG$ for some $3{\times}3$ matrix A.

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References

  1. L. J. Alias and N. Gurbuz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113-127.
  2. C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), no. 3, 355-359. https://doi.org/10.1017/S0017089500008946
  3. C. Baikoussis and T. Koufogiorgos, Helicoidal surfaces with prescribed mean or Gaussian curvature, J. Geom. 63 (1998), no. 1-2, 25-29. https://doi.org/10.1007/BF01221235
  4. C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Mat. Messina Ser. II 2(16) (1993), 31-42.
  5. B.-Y. Chen and M. Petrovic, On spectral decomposition of immersions of finite type, Bull. Austral. Math. Soc. 44 (1991), no. 1, 117-129. https://doi.org/10.1017/S0004972700029518
  6. B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186. https://doi.org/10.1017/S0004972700013162
  7. S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204. https://doi.org/10.1007/BF01425237
  8. M. Choi, D.-S. Kim, and Y. H. Kim, Helicoidal surfaces with pointwise 1-type Gauss map, J. Korean Math. Soc. 46 (2009), no. 1, 215-223. https://doi.org/10.4134/JKMS.2009.46.1.215
  9. M. Choi, D.-S. Kim, Y. H. Kim, and D. W. Yoon, Circular cone and its Gauss map, Colloq. Math. 129 (2012), no. 2, 203-210. https://doi.org/10.4064/cm129-2-4
  10. S. M. Choi, On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), no. 2, 351-367. https://doi.org/10.21099/tkbjm/1496162874
  11. S. M. Choi, On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space, Tsukuba J. Math. 19 (1995), no. 2, 285-304. https://doi.org/10.21099/tkbjm/1496162870
  12. F. Dillen, J. Pas, and L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18 (1990), no. 3, 239-246.
  13. M. P. do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J. (2) 34 (1982), no. 3, 425-435. https://doi.org/10.2748/tmj/1178229204
  14. U. Dursun, Flat surfaces in the Euclidean space $\mathbb{E}^3$ with pointwise 1-type Gauss map, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 469-478.
  15. T. Hasanis and T. Vlachos, Hypersurfaces of $E_{n+1}\;satisfying\;{\Delta}x\;=\;Ax+B$, J. Austral. Math. Soc. Ser. A 53 (1992), no. 3, 377-384. https://doi.org/10.1017/S1446788700036545
  16. U.-H. Ki, D.-S. Kim, Y. H. Kim, and Y.-M. Roh, Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space, Taiwanese J. Math. 13 (2009), no. 1, 317-338. https://doi.org/10.11650/twjm/1500405286
  17. D.-S. Kim, On the Gauss map of quadric hypersurfaces, J. Korean Math. Soc. 31 (1994), no. 3, 429-437.
  18. D.-S. Kim, On the Gauss map of hypersurfaces in the space form, J. Korean Math. Soc. 32 (1995), no. 3, 509-518.
  19. D.-S. Kim, J. R. Kim, and Y. H. Kim, Cheng-Yau operator and Gauss map of surfaces of revolution, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1319-1327. https://doi.org/10.1007/s40840-015-0234-x
  20. D.-S. Kim and Y. H. Kim, Surfaces with planar lines of curvature, Honam Math. J. 32 (2010), no. 4, 777-790. https://doi.org/10.5831/HMJ.2010.32.4.777
  21. D.-S. Kim, Y. H. Kim and D. W. Yoon, Extended B-scrolls and their Gauss maps, Indian J. Pure Appl. Math. 33 (2002), no. 7, 1031-1040.
  22. D.-S. Kim and B. Song, On the Gauss map of generalized slant cylindrical surfaces, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 20 (2013), no. 3, 149-158.
  23. Y. H. Kim and N. C. Turgay, Surfaces in $\mathbb{E}^3$ with $L_1$-pointwise 1-type Gauss map, Bull. Korean Math. Soc. 50 (2013), no. 3, 935-949. https://doi.org/10.4134/BKMS.2013.50.3.935
  24. T. Levi-Civita, Famiglie di superficie isoparametrische nell'ordinario spacio euclideo, Atti. Accad. naz Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 26 (1937), 355-362.
  25. E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569-573. https://doi.org/10.1090/S0002-9947-1970-0259768-5