Jih-Hsin Cheng

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.

Uniformization of spherical CR manifolds

Let

The Harnack Estimate for the Yamabe Flow on CR Manifolds of Dimension 3

M

Chain-preserving diffeomorphisms and CR equivalence

be a closed (compact with no boundary) spherical

An integral formula on the scalar curvature of algebraic manifolds

CR

Connected Sum of Spherical CR Manifolds with Positive CR Yamabe Constant

manifold of dimension

Minimal surfaces in pseudohermitian geometry

2n+1

Existence and uniqueness for P-area minimizers in the Heisenberg group

. Let

REGULARITY OF C 1 SMOOTH SURFACES WITH PRESCRIBED p-MEAN CURVATURE IN THE HEISENBERG GROUP

\widetilde{M}