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

TL;DR

This paper explores the relationship between surfaces in Half-Pipe space and divergence-free vector fields on the hyperbolic plane, establishing new links with harmonic Lagrangian vector fields and infinitesimal Teichmüller theory.

Contribution

It introduces a correspondence between mean surfaces in Half-Pipe space and harmonic Lagrangian vector fields, including existence and uniqueness results for their extensions.

Findings

Infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field.

Established existence and uniqueness of harmonic Lagrangian extensions.

Characterized Zygmund and little Zygmund conditions with bounds.

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.