  This paper investigates the hanging chain problem in the simply isotropic
plane as well as its 2-dimensional analog in the simply isotropic space. The
simply isotropic plane and space are two- and three-dimensional geometries
equipped with a degenerate metric whose kernel has dimension 1. Although the
metric is degenerate, the hanging chain and hanging surface problems are
well-posed if we employ the relative arc length and relative area to measure
the weight. Here, the concepts of relative arc length and relative area emerge
by seeing the simply isotropic geometry as a relative geometry. In addition to
characterizing the simply isotropic catenary, i.e., the solutions of the
hanging chain problem, we also prove that it is the generating curve of a
minimal surface of revolution in the simply isotropic space. Finally, we obtain
the 2-dimensional analog of the catenary, the so-called singular minimal
surfaces, and determine the shape of a hanging surface of revolution in the
simply isotropic space.
