  A noncommutative algebra corresponding to the classical catenoid is
introduced together with a differential calculus of derivations. We prove that
there exists a unique metric and torsion-free connection that is compatible
with the complex structure, and the curvature is explicitly calculated. A
noncommutative analogue of the fact that the catenoid is a minimal surface is
studied by constructing a Laplace operator from the connection and showing that
the embedding coordinates are harmonic. Furthermore, an integral is defined and
the total curvature is computed. Finally, classes of left and right modules are
introduced together with constant curvature connections, and bimodule
compatibility conditions are discussed in detail.
