A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk.We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.  2005 Elsevier Inc. All rights reserved