Scherk surface
Encyclopedia
In mathematics
, a Scherk surface (named after Heinrich Scherk
) is an example of a minimal surface
. Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphism
s of hyperbolic space
.
Consider the following minimal surface problem on a square in the Euclidean plane: for a natural number
n, find a minimal surface Σn as the graph of some function
such that
That is, un 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
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Scherk surface (named after Heinrich Scherk
Heinrich Scherk
Heinrich Ferdinand Scherk was a German mathematician notable for his work on minimal surfaces and the distribution of prime numbers. He is also notable as the doctoral advisor of Ernst Kummer.-External links: **...
) is an example of a minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
. Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
s of hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
.
Construction of a simple Scherk surface
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n, find a minimal surface Σn as the graph of some function
such that
That is, un 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
More general Scherk surfaces
One can consider similar minimal surface problems on other quadrilateralQuadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
s in the Euclidean plane. One can also consider the same problem on quadrilaterals in the hyperbolic plane
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
. 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.