Jenn-Fang Hwang

Properly embedded and immersed minimal surfaces in the Heisenberg group

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.

A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group

Abstract In this paper, we study the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group ℍ1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ1.

Growth property for the minimal surface equation in unbounded domains

Here we prove that if u satisfies the minimal surface equation in an unbounded domain Q which is properly contained in a half plane, then the growth rate of u is of the same order as the shape of Q and ulan .

Phragmén-Lindelöf theorem for the minimal surface equation

It is proved that if u satisfies the minimal surface equation in an unbounded domain n which is properly contained in a half plane, then the growth property of u depends on 0 and the boundary value of u only.

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A ⊂ R3 is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perrons method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffmans first theorem is not generalizable without further qualification.