Hermann Karcher's

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The mobile collection is a set of five minimal surfaces with rather diverse properties and history. It is a random collection of surfaces with close connection to Hermann Karcher's work, each surface selected by mathematical as well as aesthetical criteria. The surfaces were numerically created in Berkeley, Erlangen and Freiburg, and manufactured by a company in East-Berlin using stereolithography technique.

See a video of the complete mobile or just its shadows. Here is a copy of this page with plain computer graphics.

See the saddle tower in a video. |
Less-Symmetric Scherk Saddle Tower (image
top left)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. |

See the final model in a video. |
Chen-Gackstatter-Karcher-Thayer SurfaceChen
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. |

See the final model in a video. |
Lawson Surface of Genus 4Lawson
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. |

See the final model in a video. |
Neovius Surface with Additional HandlesIn
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. |

See the final model in a video. |
Hoffman-Karcher-Wei HelicoidThe 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: 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. |

Author:
Konrad Polthier, Berlin, Germany

Homepage H. Karcher