- Scherk surface
In
mathematics , a Scherk surface is an example of aminimal surface . Sherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonicdiffeomorphism s ofhyperbolic space .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 Σ"n" as the graph of some function:u_{n} : left( - frac{pi}{2}, + frac{pi}{2} ight) imes left( - frac{pi}{2}, + frac{pi}{2} ight) o mathbb{R}
such that
:lim_{y o pm pi / 2} u_{n} left( x, y ight) = + n mbox{ for } - frac{pi}{2} < x < + frac{pi}{2},:lim_{x o pm pi / 2} u_{n} left( x, y ight) = - n mbox{ for } - frac{pi}{2} < y < + frac{pi}{2}.
That is, "u""n" satisfies the
minimal surface equation :mathrm{div} left( frac{ abla u_{n} (x, y)}{sqrt{1 + | abla u_{n} (x, y) |^{2} ight) equiv 0
and
:Sigma_{n} = left{ (x, y, u_{n}(x, y)) in mathbb{R}^{3} left| - frac{pi}{2} < x, y < + frac{pi}{2} ight. ight}.
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{pi}{2}, + frac{pi}{2} ight) imes left( - frac{pi}{2}, + frac{pi}{2} ight) o mathbb{R},:u(x, y) = log left( frac{cos (x)}{cos (y)} ight).
That is, the Scherk surface over the square is
:Sigma = left{ left. left(x, y, log left( frac{cos (x)}{cos (y)} ight) ight) in mathbb{R}^{3} ight| - frac{pi}{2} < x, y < + frac{pi}{2} ight}.
More general Scherk surfaces
One can consider similar minimal surface problems on other
quadrilateral s 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 Sherk 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 .External links
* [http://eom.springer.de/S/s083350.htm Scherk surface] at the [http://eom.springer.de/default.htm Springer Online Encyclopaedia of Mathematics]
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