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

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.