Björling problem

{{Short description|Problem in differential geometry}}

File:Catalan's Minimal Surface.png

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 Gabriel Björling,E.G. Björling, Arch. Grunert, IV (1844) pp. 290 with further refinement by Hermann Schwarz.H.A. Schwarz, J. reine angew. Math. 80 280-300 1875

The problem can be solved by extending the surface from the curve using complex analytic continuation. If $c(s)$ is a real analytic curve in $\backslash mathbb\{R\}^3$ defined over an interval I, with $c\text{'}(s)\backslash neq\; 0$ and a vector field $n(s)$ along c such that $||n(t)||=1$ and $c\text{'}(t)\backslash cdot\; n(t)=0$, then the following surface is minimal:

:$X(u,v)\; =\; \backslash Re\; \backslash left\; (\; c(w)\; -\; i\; \backslash int\_\{w\_0\}^w\; n(w)\backslash times\; c\text{'}(w)\; \backslash ,\; dw\; \backslash right)$

where $w\; =\; u+iv\; \backslash in\; \backslash Omega$, $u\_0\backslash in\; I$, and $I\; \backslash subset\; \backslash Omega$ is a simply connected domain where the interval is included and the power series expansions of $c(s)$ and $n(s)$ are convergent.Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C³: 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

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.{{cite journal |author=W.H. Meeks III |title=The classification of complete minimal surfaces in R³ with total curvature greater than $-8\backslash pi$ |journal=Duke Math. J. |volume=48 |year=1981 |pages=523–535 |issue=3 |doi=10.1215/S0012-7094-81-04829-8 |mr=630583 |zbl=0472.53010}}

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.Björling problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bj%C3%B6rling_problem&oldid=23196