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Properly embedded and immersed minimal surfaces in the Heisenberg group

  • Jih-Hsin Cheng (a1) and Jenn-Fang Hwang (a2)

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We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We givn an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

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References

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[1]Cheng, J.-H., Hwang, J.-F., Malchiodi, A. and Yang, P., ‘Minimal surfaces in pseudohermitian geometry’, arXiv: math.DG/0401136.
[2]Collin, P., ‘Topologie et courbure des surfaces minimales de R 3, Ann. of Math. (2) 145 (1997), 131.
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Properly embedded and immersed minimal surfaces in the Heisenberg group

  • Jih-Hsin Cheng (a1) and Jenn-Fang Hwang (a2)

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