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

TL;DR

This paper introduces a Codazzi-like equation for $C^{1}$ surfaces in the Heisenberg group, enabling analysis of the singular set's structure, size, and regularity, with implications for understanding surface geometry in sub-Riemannian spaces.

Contribution

It establishes a new Codazzi-like equation for $p$-area elements and uses it to analyze the singular set's local and global structure in the Heisenberg group.

Findings

Derived a Codazzi-like differential equation for characteristic curves.

Estimated the size and regularity of the singular set.

Proved a fundamental theorem for local surfaces in $oldsymbol{H}_{1}$.

Abstract

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 $\boldsymbol{H}_{1}$.