Properly embedded and immersed minimal surfaces in the Heisenberg group

Abstract

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.