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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: [email protected]
Jenn-Fang Hwang
Affiliation:
Institute of Mathematics, Academia Sinica, Nankang, Taipei, Taiwan, 11529, Republic of China, e-mail: [email protected]
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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

[1]Cheng, J.-H., Hwang, J.-F., Malchiodi, A. and Yang, P., ‘Minimal surfaces in pseudohermitian geometry’, arXiv: math.DG/0401136.Google Scholar
[2]Collin, P., ‘Topologie et courbure des surfaces minimales de R 3, Ann. of Math. (2) 145 (1997), 1–31.CrossRefGoogle Scholar
[3]Costa, C., ‘Example of a complete minimal immersion in R 3 of genus one and three embedded ends’, Bol. Soc. Brasil. Mat. 15 (1984), 47–54.CrossRefGoogle Scholar
[4]Dierkes, U., Hildebrandt, S., Küster, A. and Wohlrob, O., Minimal surfaces I, II, Grundelehren der Mathematischen Wissenschaften 295, 296 (Springer-Verlag, Berlin, 1992).Google Scholar
[5]Fang, Y., Lectures on Minimal surfaces in R3, Proceedings of Centre for Mathematics and its Applications 35 (Australian National University, Canberra, Australia, 1996).Google Scholar
[6]Garofalo, N. and Pauls, S., ‘The Bernstein problem in the Heisenberg group’, arXiv:math.DG/0209065.Google Scholar
[7]Garofalo, N. and Pauls, S.. ‘The Bernstein problem in the Heisenberg group’, (preprint).Google Scholar
[8]Hoffman, D., Karcher, H. and Wei, F., ‘The genus one helicoid and the minimal surfaces that led to its discovery’, in Global Analysis and Moderen Mathematics, (Uhlenbeck, K., Editor) (Publish or Perish Press, 1993), pp. 119–170.Google Scholar
[9]Hoffman, D. and Meeks, W.H. III, ‘Complete embedded minimal surfaces of finite total curvature’, Bull. Amer. Math. Soc. 12 (1985), 134–136.CrossRefGoogle Scholar
[10]Hoffman, D. and Meeks, W.H. III, ‘The strong halfspace theorem for minimal surfaces’, Invent. Math. 101 (1990), 373–377.CrossRefGoogle Scholar
[11]Meeks, W. H. III and Rosenberg, H., ‘The uniquences of the helicoid and the asymptotic geometry of properly empbedded minimal surfaces with finite topology’, (preprint).Google Scholar
[12]Monge, G., Application de l'analyse à la géométrie (Bachelier, Paris, 1850).Google Scholar
[13]Nitsche, J.C.C., Vorlesungen über Minimalflächen, Grundelehren der Math. 199 (Springer-Verlag, Berlin, New York, 1975).CrossRefGoogle Scholar
[14]Osserman, R., A survey of minimal surfaces (Dover Publications Inc., New York, 1986).Google Scholar
[15]Pauls, S.D., ‘Minimal surfaces in the Heisenberg group’, Geom. Dedicata 104 (2004), 201–231.CrossRefGoogle Scholar