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In mathematics, a Scherk surface is an example of a minimal surface. Sherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space.
Additional recommended knowledge
Construction of a simple Scherk surface
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
That is, un satisfies the minimal surface equation
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 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 Sherk 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.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Scherk_surface". A list of authors is available in Wikipedia.|