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

TL;DR

This paper proves that in the Heisenberg group, surfaces with prescribed p-mean curvature have characteristic curves that are twice differentiable, and the Legendrian normal gains an additional derivative, leading to smoother seed curves.

Contribution

It establishes the regularity of characteristic and seed curves for C^{1} surfaces with prescribed p-mean curvature in the Heisenberg group, introducing characteristic coordinates and jump formulas.

Findings

Characteristic curves are C^{2} smooth with curvature -H.

Legendrian normal gains one derivative, becoming C^{2} smooth.

Uniqueness of characteristic and seed curves passing through a point.

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.