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/1963/6556