Harold Rosenberg

A classical theorem of Bernstein states that the only entire minimal graph over the euclidean plane E of dimension 2, is a plane. A harmonic map from the unit disk H to the plane E, is a map whose coordinate functions are harmonic functions on the disk. In the 1950’s, Heinz gave a proof of Bernsteins’ theorem by first proving there is no harmonic diffeomorphism from H onto E.

We will discuss graphs over H that are minimal surfaces ( in HxR, where H has the hyperbolic metric ). When the graph is entire (defined over all of H), the vertical projection to H is a harmonic diffeomorphism of the graph onto H ; the notion of harmonicity depends on the hyperbolic metric.

We will show how to construct entire minimal graphs over H that are conformally the complex plane C. Then the vertical projection yields a harmonic diffeomorphism from C onto H. This settles (negatively) a conjecture of R Schoen, stating that no such harmonic diffeomorphism exists.