In his 1835 paper, Heinrich Ferdinand Scherk introduces five new minimal surfaces. The first is his family of doubly periodic surfaces, and the fifth he describes by the equation

sin(z) = sinh(x)\cdot \sinh(y).

It is, in fact, the conjugate of his first new surface, the doubly periodic one with parallel ends.

Scherk only describes the orthogonal case. These surfaces were tremendously influential, comparable only to the Costa surface.

From far away, these surfaces look like two intersecting planes. It is a famous open problem to show that any complete, embedded minimal surface that is asymptotic to two intersecting planes is one the Scherk surfaces.This is only known to be true in the periodic case by work of William Meeks and Michael Wolf.



W. Meeks, M. Wolf: Minimal surfaces with the area growth of two planes; the case of infinite symmetry

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