Minimal Graphs in H × R and Their Projections

June 29, 2007 There has been a recent resurgence of interest in the subject of minimal surfaces in product three-manifolds, for example H ×R [Ros02], [MR04], [MIR05], [NR02], [Dan06], [FM05], [HET05]. The purpose of this note is to describe a construction and some examples originating in the harmonic maps literature whose consequences for minimal surfaces seem to have escaped notice. We organize this note as follows. First we recall how harmonic maps to H (or a Riemannian complex) lead, via associated maps to real trees, to minimal surfaces in H × R. This occupies the first three sections and leads to a proposition describing minimal graphs in terms of harmonic maps to trees and their folds. In the next section, we apply the construction to some examples from the harmonic maps literature to create some minimal graphs whose shapes have not yet been explored much. We describe rapidly oscillating graphs, including one derived from a harmonic map to the Cayley graph of a free group. In terms of asymptotic behavior, we describe a non-trivial graph whose boundary values are constant except at a single point, and graphs which accumulate only over a Cantor set on S ∞. We relate Jenkins-Serrin constructions to trees of finite total valence in section five, and conclude in section six with a non-orientable example (over the product of a surface with RP) corresponding to a harmonic map to a tree which does not fold. The author is grateful to Bill Meeks and Matthias Weber for several stimulating and encouraging conversations.