Abstract
Recently, K. T. Hahn introduced a new pseudodifferential metric for complex manifolds by means of an extremal problem. This metric is investigated in the special case of Riemann surfaces. First, the metric is related to two other extremal problems on a Riemann surface. Next, basic properties of the Hahn metric are studied; in particular, it is shown that the Hahn metric is complete if it is nontrivial. For simply and doubly connected Riemann surfaces the Hahn metric is explicitly calculated; it is also studied on tori. Finally, the Hahn metric is shown to have generalized Gaussian curvature at least −4; for the unit disk it has constant curvature −4.
