The Karcher-Meeks-Rosenberg surfaces, short KMR surfaces, are complete, embedded doubly periodic minimal surfaces with two top and two bottom parallel annular ends in the quotient.

They were first described in 1988 by Hermann Karcher, and independently and differently by William Meeks and Harold Rosenberg in 1989. The latter description is analogous to that of a family of triply periodic minimal surfaces of genus 3 in Meeks’ thesis from 1975, of which they can be understood as limits, and I suspect that he new about them much earlier.

These surfaces come in a 3-parameter family, allowing for shearing the period lattice and tilting the ends of the most symmetric examples above. In the latter case the surfaces resemble a doubly periodic version of Riemann’s singly periodic surface.

KMR-3b

In 2005, Joaquín Pérez, Magdalena Rodríguez, and Martin Traizet proved that these are the only embedded doubly periodic minimal surfaces with parallel ends of genus one. With non-parallel ends, one has the Karcher-Scherk examples, and there are many examples of higher genus.

Resources

Mathematica Notebooks cases 1, 2, 3

PoVRay Sources 1, 2, 3

H. Karcher. Embedded minimal surfaces derived from Scherk’s examples. Manuscripta Math., 62 (83–114), 1988

W. H. Meeks III and H. Rosenberg. The global theory of doubly periodic minimal surfaces. Invent. Math., 97 (351–379), 1989

Joaquín Pérez, M. Magdalena Rodríguez, and Martin Traizet, The classification of doubly periodic minimal tori with parallel ends, J. Differential Geom. 69 (2005), 523–577