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In this paper we consider surfaces of class $C^1$ with continuous prescribedmean curvature in a three-dimensional contact sub-Riemannian manifold and provethat their characteristic curves are of class $C^2$. This regularity resultalso holds for critical points of the sub-Riemannian perimeter under a volumeconstraint. All results are valid in the first Heisenberg group $\mathbb{H}^1$.

Regularity of $C^1$ surfaces with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds