Enneper Surface

Alfred Enneper (1830-1885) was a prolific geometer. He liked to write long and technical papers, often in several parts, picking up threads from several years ago. One of his recurring topics were surfaces with special curve systems, like planar curvature lines.

One of his (many) examples is now commonly called the Enneper surface. To my knowledge, he describes it first 1871, where he derives its algebraic description from its Enneper-Weierstraß representation

$G(z)=z, \qquad dh = z\, dz$ .

Curiously, Enneper also writes down the parametrization of the conjugate surface but does not realize that they are congruent. In fact, the entire associate family of Enneper’s surface is obtained by just rotating the surface in space. This is related to the amusing fact that the Enneper surface is intrinsically a surface of revolution, i.e. its first fundamental form is rotationally symmetric.

The surface is not embedded. Its single end has, however, a limiting normal and winding number 3, which makes it one of the simplest non-embedded minimal surfaces. Besides the catenoid it is the only complete minimal surface of finite total curvature -4π.

Enneper’s initial interest in this minimal surface comes from the fact that its curvature lines are all planar. This allows to make a model of the curvature line net out of plywood (or stiff paper).