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Properly embedded and immersed minimal surfaces in the Heisenberg group

Published online by Cambridge University Press:  17 April 2009

Jih-Hsin Cheng  and
Jenn-Fang Hwang
Jih-Hsin Cheng
Affiliation:
Institute of Mathematics, Academia Sinica, Nankang, Taipei, Taiwan, 11529, Republic of China, e-mail: cheng@math.sinica.edu.tw
Jenn-Fang Hwang
Affiliation:
Institute of Mathematics, Academia Sinica, Nankang, Taipei, Taiwan, 11529, Republic of China, e-mail: majfh@ccvax.sinica.edu.tw
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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 3 , December 2004 , pp. 507 - 520
Copyright
Copyright © Australian Mathematical Society 2004

References

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