Properly embedded and immersed minimal surfaces in the Heisenberg group

Details Properly embedded and immersed minimal surfaces in the Heisenberg group Properly embedded and immersed minimal surfaces in the Heisenberg group

2004-07-07

arxiv.org/abs/math/0407094v1

Bull. Aus. Math. Soc., 70 (2004) 507-520.

Mathematics

Differential Geometry

13 pages, 3 figures

Scientific paper

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 give 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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