A class of surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ associated to harmonic functions and a relation between CMC-1/2 and flat surfaces

Abstract

We introduce a geometric motivated method to construct immersions into $\mathbb{H}^2\times\mathbb{R}$ from a smooth function $\varphi$ defined on an open set of the unit disc, and study the relation between the geometry of the immersion in terms of partial differential equations for $\varphi$. We give two applications of the method. First, we introduce the class of surfaces generated by harmonic functions and show that they have properties analogous to minimal surfaces in $\mathbb {R}^3$. We also exhibit an explicit local relation between CMC 1/2 and flat surfaces in $\mathbb{H}^2\times\mathbb{R}$.