|
|
| Line 1: |
Line 1: |
| [[File:Scherkassociatefamily.gif|thumb|Animation of Scherk's first and second surface transforming into each other: they are members of the same [[associate family]] of minimal surfaces.]]
| | The name of the writer is Figures but it's not the most masucline title out there. To do aerobics is a thing that I'm totally addicted to. Years ago we moved to North Dakota and I adore every day residing here. I am a meter reader.<br><br>Here is my web-site ... [http://www.siccus.net/blog/15356 www.siccus.net] |
| | |
| In [[mathematics]], a '''Scherk surface''' (named after [[Heinrich Scherk]]) is an example of a [[minimal surface]]. Scherk described two complete embedded minimal surfaces in 1834;<ref>H.F. Scherk, Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen, Journal für die reine und angewandte Mathematik, Volume 13 (1835) pp. 185–208 [http://books.google.co.uk/books?id=K5tGAAAAcAAJ&lpg=PA185&ots=tV5mR2nP-m&dq=%22Bemerkungen%20%C3%BCber%20die%20kleinste%20Fl%C3%A4che%20innerhalb%20gegebener%20Grenzen%22&pg=PA185#v=onepage&q&f=false]</ref> 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]] and [[helicoid]]).<ref>http://www-history.mcs.st-andrews.ac.uk/Biographies/Scherk.html</ref> The two surfaces are [[Associate family|conjugates]] of each other.
| |
| | |
| Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic [[diffeomorphism]]s of [[hyperbolic space]].
| |
| | |
| ==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===
| |
| | |
| [[Image:ScherkSurface.png|thumb|The Scherk surface Σ given by the graph of ''u''(''x'', ''y'') = log ( cos(''x'') / cos(''y'') ) for ''x'' and ''y'' between −''π''/2 and ''π''/2.]]
| |
| [[File:Superficie di scherk.jpg|thumb|Nine periods of the 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<sub>''n''</sub> as the graph of some function
| |
| | |
| :<math>u_{n} : \left( - \frac{\pi}{2}, + \frac{\pi}{2} \right) \times \left( - \frac{\pi}{2}, + \frac{\pi}{2} \right) \to \mathbb{R}</math>
| |
| | |
| such that
| |
| | |
| :<math>\lim_{y \to \pm \pi / 2} u_{n} \left( x, y \right) = + n \text{ for } - \frac{\pi}{2} < x < + \frac{\pi}{2},</math>
| |
| :<math>\lim_{x \to \pm \pi / 2} u_{n} \left( x, y \right) = - n \text{ for } - \frac{\pi}{2} < y < + \frac{\pi}{2}.</math>
| |
| | |
| That is, ''u''<sub>''n''</sub> satisfies the [[minimal surface equation]]
| |
| | |
| :<math>\mathrm{div} \left( \frac{\nabla u_{n} (x, y)}{\sqrt{1 + | \nabla u_{n} (x, y) |^{2}}} \right) \equiv 0</math>
| |
| | |
| and
| |
| | |
| :<math>\Sigma_{n} = \left\{ (x, y, u_{n}(x, y)) \in \mathbb{R}^{3} \left| - \frac{\pi}{2} < x, y < + \frac{\pi}{2} \right. \right\}.</math>
| |
| | |
| 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
| |
| | |
| :<math>u : \left( - \frac{\pi}{2}, + \frac{\pi}{2} \right) \times \left( - \frac{\pi}{2}, + \frac{\pi}{2} \right) \to \mathbb{R},</math>
| |
| :<math>u(x, y) = \log \left( \frac{\cos (x)}{\cos (y)} \right).</math>
| |
| | |
| That is, the '''Scherk surface''' over the square is
| |
| | |
| :<math>\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\}.</math>
| |
| | |
| ===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 the [[Hyperbolic space|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]].
| |
| | |
| ==Scherk's second surface==
| |
| | |
| [[File:Scherk's second surface.png|thumb|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:
| |
| :<math>\sin(z) - \sinh(x)\sinh(y)=0</math>
| |
| | |
| It has the [[Weierstrass–Enneper parameterization]]
| |
| <math>f(z) = \frac{4}{1-z^4}</math>, <math>g(z) = iz</math>
| |
| and can be parametrized as:<ref>Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd ed., CRC press 2002</ref>
| |
| | |
| :<math>x(r,\theta) = 2 \Re ( \ln(1+re^{i \theta}) - \ln(1-re^{i \theta}) ) = \ln \left( \frac{1+r^2+2r \cos \theta}{1+r^2-2r \cos \theta} \right)</math>
| |
| | |
| :<math>y(r,\theta) = \Re ( 4i \tan^{-1}(re^{i \theta})) = \frac{1+r^2-2r \sin\theta}{1+r^2+2r \sin \theta}</math>
| |
| | |
| :<math>z(r,\theta) = \Re ( 2i(-\ln(1-r^2e^{2i \theta}) + \ln(1+r^2e^{2i \theta}) ) = 2 \tan^{-1}\left( \frac{2 r^2 \sin 2\theta}{r^4-1} \right)</math>
| |
| | |
| for <math>\theta \in [0, 2\pi)</math> and <math>r \in (0,1)</math>. 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.<ref>Nikolaos Kapuoleas, Constructions of minimal surfaces by glueing minimal immersions. In Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001 p. 499</ref><ref>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)</ref> To minimize confusion it is useful to refer to it as Scherk's singly periodic surface or the Scherk-tower.
| |
| | |
| ==External links==
| |
| * {{springerEOM | id=Scherk_surface | first=I.Kh. | last=Sabitov | oldid=15282 }}
| |
| * Scherk's first surface in MSRI Geometry [http://archive.msri.org/about/sgp/jim/geom/minimal/library/scherk1/index.html]
| |
| * Scherk's second surface in MSRI Geometry [http://archive.msri.org/about/sgp/jim/geom/minimal/library/scherk2/index.html]
| |
| * Scherk's minimal surfaces in Mathworld [http://mathworld.wolfram.com/ScherksMinimalSurfaces.html]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Minimal surfaces]]
| |
| [[Category:Differential geometry]]
| |
The name of the writer is Figures but it's not the most masucline title out there. To do aerobics is a thing that I'm totally addicted to. Years ago we moved to North Dakota and I adore every day residing here. I am a meter reader.
Here is my web-site ... www.siccus.net