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2014-01-11T15:09:01Z

Electric fields

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[[File:Catalan's Minimal Surface.png|thumb|Catalan's minimal surface. It can be defined as the minimal surface symmetrically passing through a cycloid.]]

In [[differential geometry]], the '''Björling problem''' is the problem of finding a [[minimal surface]] passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by Swedish mathematician [[Emanuel Björling|Emanuel Gabriel Björling]],<ref>E.G. Björling, Arch. Grunert , IV (1844) pp. 290</ref> with further refinement by Schwarz.<ref>H.A. Schwarz, J. reine angew. Math. 80 280-300 1875</ref>

The problem can be solved by extending the surface from the curve using complex [[analytic continuation]]. If <math>c(s)</math> is a real analytic curve in ℝ<sup>3</sup> defined over an interval ''I'', with <math>c'(s)\neq 0</math> and a vector field <math>n(s)</math> along ''c'' such that <math>||n(t)||=1</math> and <math>c'(t)\cdot n(t)=0</math>, then the following surface is minimal:

:<math>X(u,v) = \Re \left ( c(w) - i \int_{w_0}^w n(w) \wedge c'(w) \, dw \right)</math>

where <math>w = u+iv \in \Omega</math>, <math>u_0\in I</math>, and <math>I \subset \Omega</math> is a simply connected domain where the interval is included and the power series expansions of <math>c(s)</math> and <math>n(s)</math> are convergent.<ref>Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in '''C'''<sup>3</sup>: Associated minimal surfaces and the Björling's problem. MIT BA Thesis. 2004 http://ocw.mit.edu/courses/mathematics/18-994-seminar-in-geometry-fall-2004/projects/main1.pdf</ref>

A classic example is [[Catalan's minimal surface]], which passes through a [[cycloid]] curve. Applying the method to a [[semicubical parabola]] produces the [[Henneberg surface]], and to a circle (with a suitably twisted normal field) a minimal [[Möbius strip]].<ref>W.H. Meeks III, The classification of complete minimal surfaces in R3 with total curvature greater than 8, Duke Math. J. 48 (1981), 523–535.</ref>

A unique solution always exists. It can be viewed as a [[Cauchy problem]] for minimal surfaces, allowing one to find a surface if a geodesic, asymptote or lines of curvature is known. In particular, if the curve is planar and geodesic, then the plane of the curve will be a symmetry plane of the surface.<ref>Björling problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bj%C3%B6rling_problem&oldid=23196</ref>

==References==

{{reflist|30em}}

==External image galleries==

* Björling Surfaces, at the Indiana Minimal Surface Archive: http://www.indiana.edu/~minimal/archive/Bjoerling/index.html

{{DEFAULTSORT:Bjorling problem}}
[[Category:Minimal surfaces]]
[[Category:Differential geometry]]

List of electromagnetism equations - Revision history

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