The Mobile Collection

The classical minimal surfaces of H.F. Scherk were found around 1835 in an attempt to solve Gergonne's problem, a boundary value problem in the cube. The Scherk surfaces were among the first candidates in Karcher's experiments to modify the Weierstraß formula of existing surfaces. He selectively increased or decreased symmetry, or twisted, or changed the topological genus by inserting new handles. The less-symmetric saddle tower originates from Scherk's saddle tower with triple symmetry and modifying the asymptotic angle of the half plane wings.

Further reading: H. Karcher Embedded Minimal Surfaces Derived from Scherk's Examples Manuscripta Math. 62, 1988.

Chen and Gackstatter discovered in 1982 surfaces of genus one and two each having an Enneper-type end of winding order three. Karcher found that the end may be generalized to have any odd winding order. Thayer constructed numerically surfaces with many more handles, i.e. higher genus, up to 35, and each can have a generalized Enneper-type end. The triply symmetric surface of the mobile has genus six and winding order 5.

Further reading: E. Thayer Higher-Genus Chen-Gackstatter Surfaces and the Weierstraß Representation for Surfaces of Infinite Genus Experimental Mathematics 4 (1) 1995.

Lawson constructs compact minimal surfaces in the 3-sphere of arbitrary genus by applying Morrey's solution of the Plateau problem in general manifolds. This work of Lawson contains a rich set of ideas among them the conjugate surface construction for minimal and constant mean curvature surfaces. Karcher elaborated and perfected the conjugate surface construction to allow the construction of a large number of new minimal and constant mean curvature surfaces in different space forms.

Further reading: H.B. Lawson Complete Minimal Surfaces in S³ Annals of Math. 92, 1970.

In the last century H. A. Schwarz and his pupil E. Neovius were among the first to specifically design new triply periodic minimal surfaces using complex analysis and the Weierstraß representation formula. The physicist A. Schoen found many more triply periodic surfaces in crystallographic cells. Karcher elaborated the conjugate surface construction to proof existence of Schoen's surfaces, and many new examples. Karcher's modification of Neovius' surface was numerically continued by Oberknapp to add a wealth of handles.

Further reading: H. Karcher The triply periodic minimal surfaces of A. Schoen and their constant mean curvature companions Manuscripta Math. 64, 1989.

The Genus-One Helicoid is a minimally embedded torus with one end and infinite total curvature. More than 200 hundred years after the helicoid of Meusnier a new embedded minimal surface with finite topology and infinite total curvature was found in 1993. Crucial to their new discovery was the characterization of the Gauss maps' essential singularity at the end of the helicoid. It is known that the initiative to the genus-one helicoid is due to Harold Rosenberg: "Hermann, why don't David and you sit down and construct such an example?"

Further reading: D. Hoffman, H. Karcher, F. Wei The Genus One Helicoid and the Minimal Surfaces that Led to its Discovery in: Global Analysis and Modern Mathematics (edited by K. Uhlenbeck), Publish and Perish, 1993.