Scherk surface

In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other.

Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space.

Scherk's first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet near z = 0 in a checkerboard pattern of bridging arches. It contains an infinite number of straight vertical lines.

Consider the following minimal surface problem on a square in the Euclidean plane: for a natural number n, find a minimal surface Σ_n as the graph of some function

such that

That is, u_n satisfies the minimal surface equation

and

What, if anything, is the limiting surface as n tends to infinity? The answer was given by H. Scherk in 1834: the limiting surface Σ is the graph of

That is, the Scherk surface over the square is

One can consider similar minimal surface problems on other quadrilaterals in the Euclidean plane. One can also consider the same problem on quadrilaterals in the hyperbolic plane. In 2006, Harold Rosenberg and Pascal Collin used hyperbolic Scherk surfaces to construct a harmonic diffeomorphism from the complex plane onto the hyperbolic plane (the unit disc with the hyperbolic metric), thereby disproving the Schoen–Yau conjecture.

Scherk's second surface looks globally like two orthogonal planes whose intersection consists of a sequence of tunnels in alternating directions. Its intersections with horizontal planes consists of alternating hyperbolas.

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