The prescribed p-mean curvature equation of low regularity in the Heisenberg group

Abstract

This work reports on the author’s recent study about regularity and the singular set of a C ¹ smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. As a differential equation, this is a degenerate hyperbolic and elliptic PDE of second order, arising from the study of CR geometry. Assuming only the p-mean curvature H ∈ C ⁰, it is shown that any characteristic curve is C ² smooth and its (line) curvature equals −H. By introducing special coordinates and invoking the jump formulas along characteristic curves, it is proved that the Legendrian (horizontal) normal gains one more derivative. Therefore the seed curves are C ² smooth. This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. In an on-going project, it is shown that the p-area element is in fact C ² smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order. Moreover, this ODE is analyzed to study the singular set.