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

Abstract

We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ 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 $C^{2}$ 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.