01152nas a2200145   4500008004100000245009700041210006900138260001000207520067600217100002200893700001500915700002000930700002000950856003600970       2012                        en  d00aA Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.0 aCodazzilike equation and the singular set for C1 smooth surfaces  bSISSA3 aIn this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. 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 ℍ 11 aMalchiodi, Andrea1 aYang, Paul1 aCheng, Jih-Hsin1 aHwang, JennFang  uhttp://hdl.handle.net/1963/655601933nas a2200145   4500008004100000245004900041210004900090260002900139520150600168100002001674700002001694700002201714700001501736856003601751       2005                        en  d00aMinimal surfaces in pseudohermitian geometry0 aMinimal surfaces in pseudohermitian geometry  bScuola Normale Superiore3 aWe consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a
characteristic curve, and as a quantity in terms of calibration geometry.
  As a differential equation, the p-minimal surface equation is degenerate
(hyperbolic and elliptic). To analyze the singular set, we formulate some {\em
extension} theorems, which describe how the characteristic curves meet the
singular set. This allows us to classify the entire solutions to this equation
and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the
Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space
$R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings
generated by Legendrian lines. We also prove a uniqueness theorem for the
Dirichlet problem under a condition on the size of the singular set in two
dimensions, and generalize to higher dimensions without any size control
condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.1 aCheng, Jih-Hsin1 aHwang, JennFang1 aMalchiodi, Andrea1 aYang, Paul  uhttp://hdl.handle.net/1963/4579