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Scherk surface

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.

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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

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

such that

$\lim_{y \to \pm \pi / 2} u_{n} \left( x, y \right) = + n \mbox{ for } - \frac{\pi}{2} < x < + \frac{\pi}{2},$

$\lim_{x \to \pm \pi / 2} u_{n} \left( x, y \right) = - n \mbox{ for } - \frac{\pi}{2} < y < + \frac{\pi}{2}.$

That is, u_n satisfies the minimal surface equation

$\mathrm{div} \left( \frac{\nabla u_{n} (x, y)}{\sqrt{1 + | \nabla u_{n} (x, y) |^{2}}} \right) \equiv 0$

and

$\Sigma_{n} = \left\{ (x, y, u_{n}(x, y)) \in \mathbb{R}^{3} \left| - \frac{\pi}{2} < x, y < + \frac{\pi}{2} \right. \right\}.$

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

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

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

That is, the Scherk surface over the square is

$\Sigma = \left\{ \left. \left(x, y, \log \left( \frac{\cos (x)}{\cos (y)} \right) \right) \in \mathbb{R}^{3} \right| - \frac{\pi}{2} < x, y < + \frac{\pi}{2} \right\}.$

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.

Category: Minimal surfaces