A Codazzi-like equation and the singular set for $C^{1}$ smooth surfaces in the Heisenberg group

In this paper, we study the structure of the singular set for a $C^{1}$ smooth surface in the $3$-dimensional Heisenberg group $\boldsymbol{H}_{1}$. 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 that Codazzi-like equation by proving a fundamental theorem for local surfaces in $\boldsymbol{H}_{1}$.