Mean surfaces in Half-Pipe space and infinitesimal Teichmüller theory

Abstract

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.

44 pages, 1 figure