Regularity of C ¹ smooth surfaces with prescribed p-mean curvature in the Heisenberg group

We consider a C¹ smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature H Ie{cyrillic, ukrainian} C⁰ we show that any characteristic curve is C² smooth and its (line) curvature equals - H in the nonsingular domain. By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are C² smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations.