We study a correspondence between smooth spacelike surfaces in the half-pipe space ℍℙ 3 and divergence-free vector fields on the hyperbolic plane ℍ 2 . We show that a particular case involves harmonic Lagrangian vector fields on ℍ 2 , which are related to mean surfaces in ℍℙ 3 . Consequently, we prove that the infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field that corresponds to a mean surface in ℍℙ 3 with prescribed boundary data at infinity. We establish both existence and, under certain assumptions, uniqueness results for harmonic Lagrangian extension of a vector field on the circle. Finally, we characterize the Zygmund and little Zygmund conditions and provide quantitative bounds in terms of the half-pipe width.
Farid Diaf (Mon,) studied this question.