Abstract
We consider the set of minimum points of the density of the hyperbolic metric for domains in the plane. We show that there is a simply connected domain whose density has a prescribed number of minimum points. Also, there is a simply connected domain for which the set of minimum points of the density is the set of integers. We also consider the level sets of the hyperbolic density. When the domain is convex, each level set bounds a convex domain. When the domain is convex and unbounded, the level sets also are unbounded and, except for a strip, the density fails to have a minimum.
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