# Regularity of C^{1} smooth surfaces with prescribed p-mean curvature in the Heisenberg group - Mathematics > Differential Geometry

Abstract: We consider a $C^{1}$ smooth surface with prescribed $p$or $H$-meancurvature 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 nonsingulardomain$.$ By introducing characteristic coordinates and invoking the jumpformulas along characteristic curves, we can prove that the Legendrian orhorizontal normal gains one more derivative. Therefore the seed curves are$C^{2}$ smooth. We also obtain the uniqueness of characteristic and seed curvespassing through a common point under some mild conditions, respectively. Theseresults can be applied to more general situations.

Author: Jih-Hsin Cheng, Jenn-Fang Hwang, Paul Yang

Source: https://arxiv.org/