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In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
. 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 In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describe ...
and
helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
). 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
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s of
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
.


Scherk's first surface

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.


Construction of a simple Scherk surface

Consider the following minimal surface problem on a square in the Euclidean plane: for a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
''n'', find a minimal surface Σ''n'' as the graph of some function :u_ : \left( - \frac, + \frac \right) \times \left( - \frac, + \frac \right) \to \mathbb such that :\lim_ u_ \left( x, y \right) = + n \text - \frac < x < + \frac, :\lim_ u_ \left( x, y \right) = - n \text - \frac < y < + \frac. That is, ''u''''n'' satisfies the minimal surface equation :\mathrm \left( \frac \right) \equiv 0 and :\Sigma_ = \left\. 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 :u : \left( - \frac, + \frac \right) \times \left( - \frac, + \frac \right) \to \mathbb, :u(x, y) = \log \left( \frac \right). That is, the Scherk surface over the square is :\Sigma = \left\.


More general Scherk surfaces

One can consider similar minimal surface problems on other
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
s in the Euclidean plane. One can also consider the same problem on quadrilaterals in the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
. 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

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: :\sin(z) - \sinh(x)\sinh(y)=0 It has the
Weierstrass–Enneper parameterization In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let f and g be functions on either ...
f(z) = \frac, g(z) = iz and can be parametrized as: :x(r,\theta) = 2 \Re ( \ln(1+re^) - \ln(1-re^) ) = \ln \left( \frac \right) :y(r,\theta) = \Re ( 4i \tan^(re^)) = \ln \left( \frac \right) :z(r,\theta) = \Re ( 2i(-\ln(1-r^2e^) + \ln(1+r^2e^) ) = 2 \tan^\left( \frac \right) for \theta \in [0, 2\pi) and 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 the saddle tower family of periodic minimal surfaces. Somewhat confusingly, this surface is occasionally called Scherk's fifth surface in the literature.David Hoffman and William H. Meeks, Limits of minimal surfaces and Scherk's Fifth Surface, Archive for rational mechanics and analysis, Volume 111, Number 2 (1990) To minimize confusion it is useful to refer to it as Scherk's singly periodic surface or the Scherk-tower.


External links

* * Scherk's first surface in MSRI Geometr

* Scherk's second surface in MSRI Geometr

* Scherk's minimal surfaces in Mathworl


References

{{Minimal surfaces Minimal surfaces Differential geometry