Mean surfaces in Half-Pipe space and infinitesimal Teichm\"uller theory

We study a correspondence between smooth spacelike surfaces in Half-Pipe space $\mathbb{HP}^3$ and divergence-free vector fields on the hyperbolic plane $\mathbb{H}^2$. We show that a particular case involves harmonic Lagrangian vector fields on $\mathbb{H}^2$, which are related to mean surfaces in $\mathbb{HP}^3$. Consequently, we prove that the infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field that corresponds to a mean surface in $\mathbb{HP}^3$ 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.