Inmathematics, aScherk surface (named afterHeinrich Scherk) is an example of aminimal surface. Scherk described two complete embedded minimal surfaces in 1834;^[1] 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 thecatenoid andhelicoid).^[2] The two surfaces areconjugates of each other.

Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonicdiffeomorphisms ofhyperbolic space.

Scherk's first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet nearz = 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 anatural numbern, find a minimal surface Σ_n as the graph of some function

${\displaystyle u_n:\left(-\frac{\pi }{2},+\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},+\frac{\pi }{2}\right)\to \mathbb{R}}$

such that

That is,u_n satisfies theminimal surface equation

${\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\mathrm{\nabla }{u}_n(x,y)}{\sqrt{1+|\mathrm{\nabla }{u}_n(x,y){|}^2}}\right)\equiv 0}$

and

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

${\displaystyle u:\left(-\frac{\pi }{2},+\frac{\pi }{2}\right)\times \left(-\frac{\pi }{2},+\frac{\pi }{2}\right)\to \mathbb{R},}$

${\displaystyle u(x,y)=\log \left(\frac{\cos (x)}{\cos (y)}\right).}$

That is, theScherk surface over the square is

One can consider similar minimal surface problems on otherquadrilaterals in the Euclidean plane. One can also consider the same problem on quadrilaterals in thehyperbolic 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 theSchoen–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.

It has implicit equation:

${\displaystyle \sin (z)-\sinh (x)\sinh (y)=0}$

It has theWeierstrass–Enneper parameterization${\displaystyle f(z)=\frac{4}{1-z^4}}$,${\displaystyle g(z)=iz}$and can be parametrized as:^[3]

${\displaystyle x(r,\theta )=2\mathrm{\Re }(\ln (1+r{e}^{i\theta })-\ln (1-r{e}^{i\theta }))=\ln \left(\frac{1+r^2+2r\cos \theta }{1+r^2-2r\cos \theta }\right)}$

${\displaystyle y(r,\theta )=\mathrm{\Re }(4i{\tan }^{-1}(r{e}^{i\theta }))=\ln \left(\frac{1+r^2-2r\sin \theta }{1+r^2+2r\sin \theta }\right)}$

${\displaystyle z(r,\theta )=\mathrm{\Re }(2i(-\ln (1-r^2{e}^{2i\theta })+\ln (1+r^2{e}^{2i\theta }))=2{\tan }^{-1}\left(\frac{2r^2\sin 2\theta }{r^4-1}\right)}$

for${\displaystyle \theta \in [0,2\pi )}$ and${\displaystyle r\in (0,1)}$. This gives one period of the surface, which can then be extended in the z-direction by symmetry.

The surface has been generalised by H. Karcher into thesaddle tower family of periodic minimal surfaces.

Somewhat confusingly, this surface is occasionally called Scherk's fifth surface in the literature.^[4][5] To minimize confusion it is useful to refer to it as Scherk's singly periodic surface or the Scherk-tower.

• Sabitov, I.Kh. (2001) [1994],"Scherk surface",Encyclopedia of Mathematics,EMS Press
• Scherk's first surface in MSRI Geometry[2]
• Scherk's second surface in MSRI Geometry[3]
• Scherk's minimal surfaces in Mathworld[4]

• Minimal surfaces
• Differential geometry

• Articles with short description
• Short description matches Wikidata