Björling problem explained

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,^[1] with further refinement by Hermann Schwarz.^[2]

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

R³

defined over an interval I, with

c'(s) ≠ 0

and a vector field

n(s)

along c such that

||n(t)||=1

and

c'(t) ⋅ n(t)=0

, then the following surface is minimal:

X(u,v)=\Re\left(c(w)-i

	w
\int
	w₀

n(w) x c'(w)dw\right)

where

w=u+iv\in\Omega

u_0\inI

, and

I\subset\Omega

is a simply connected domain where the interval is included and the power series expansions of

c(s)

and

n(s)

are convergent.^[3]

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.^[4]

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.^[5]

External image galleries

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

Notes and References

E.G. Björling, Arch. Grunert, IV (1844) pp. 290
H.A. Schwarz, J. reine angew. Math. 80 280-300 1875
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
W.H. Meeks III . The classification of complete minimal surfaces in R³ with total curvature greater than
-8\pi

. Duke Math. J. . 48 . 1981 . 523–535 . 3 . 10.1215/S0012-7094-81-04829-8 . 630583 . 0472.53010.
Björling problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bj%C3%B6rling_problem&oldid=23196