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HELICOIDAL MINIMAL SURFACES IN ℍ2×ℝ

  • YOUNG WOOK KIM (a1), SUNG-EUN KOH (a2), HEAYONG SHIN (a3) and SEONG-DEOG YANG (a4)

Abstract

It is shown that a minimal surface in ℍ2×ℝ is invariant under a one-parameter group of screw motions if and only if it lies in the associate family of helicoids. It is also shown that the conjugate surfaces of the parabolic and hyperbolic helicoids in ℍ2×ℝ are certain types of catenoids.

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Copyright

Corresponding author

For correspondence; e-mail: skoh@konkuk.ac.kr

Footnotes

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The first named author was supported by NRF 2009-0086794. The second named author was supported by NRF 2009-0086794 and NRF 2009-0086441.

Footnotes

References

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[1]Abresch, U. and Rosenberg, H., ‘A Hopf differential for constant mean curvature surfaces in 𝕊2×ℝ and ℍ×ℝ’, Acta Math. 193 (2004), 141174.
[2]Daniel, B., ‘Isometric immersions into 𝕊n×ℝ and ℍn×ℝ and applications to minimal surfaces’, Trans. Amer. Math. Soc. 361 (2009), 62556282.
[3]Galvez, D., Martinez, A. and Mira, P., ‘The Bonnet problem for surfaces in homogeneous 3-manifolds’, Comm. Anal. Geom. 16 (2008), 907935.
[4]Hauswirth, L., Sa Earp, R. and Toubiana, E., ‘Associate and conjugate minimal immersions in 𝕄×ℝ’, Tohoku Math. J. 60 (2008), 267286.
[5]Kim, Y. W., Koh, S.-E., Shin, H. and Yang, S.-D., ‘Helicoids in 𝕊2×ℝ and ℍ2×ℝ’, Pacific J. Math. 242 (2009), 281297.
[6]Mira, P. and Pastor, A. S., ‘Helicoidal maximal surfaces in Lorentz–Minkowski space’, Monatsch. Math. 140 (2003), 315334.
[7]Meeks, W. and Rosenberg, H., ‘The theory of minimal surfaces in M×ℝ’, Comment. Math. Helv. 80 (2005), 811858.
[8]Nitsche, J. C. C., Lectures on Minimal Surfaces, Vol. 1 (Cambridge University Press, Cambridge, 2008).
[9]Rosenberg, H., ‘Minimal surfaces in M×ℝ’, Illinois J. Math. 46 (2002), 11771195.
[10]Sa Earp, R., ‘Parabolic and hyperbolic screw motion surfaces in ℍ2×ℝ’, J. Aust. Math. Soc. 85 (2008), 113143.
[11]Sa Earp, R. and Toubiana, E., ‘Screw motion surfaces in ℍ2×ℝ and 𝕊2×ℝ’, Illinois J. Math. 49 (2005), 13231362.
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Keywords

MSC classification

HELICOIDAL MINIMAL SURFACES IN ℍ2×ℝ

  • YOUNG WOOK KIM (a1), SUNG-EUN KOH (a2), HEAYONG SHIN (a3) and SEONG-DEOG YANG (a4)

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