Misplaced Pages

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, with further refinement by Hermann Schwarz.

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

${\displaystyle X(u,v)=\Re \left(c(w)-i\int _{w_{0}}^{w}n(w)\times c'(w)\,dw\right)}$

where ${\displaystyle w=u+iv\in \Omega }$, ${\displaystyle u_{0}\in I}$, and ${\displaystyle I\subset \Omega }$ is a simply connected domain where the interval is included and the power series expansions of ${\displaystyle c(s)}$ and ${\displaystyle n(s)}$ are convergent.

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.

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.

References

1. E.G. Björling, Arch. Grunert, IV (1844) pp. 290
2. H.A. Schwarz, J. reine angew. Math. 80 280-300 1875
3. Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C: 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
4. W.H. Meeks III (1981). "The classification of complete minimal surfaces in R with total curvature greater than ${\displaystyle -8\pi }$". Duke Math. J. 48 (3): 523–535. doi:10.1215/S0012-7094-81-04829-8. MR 0630583. Zbl 0472.53010.
5. Björling problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bj%C3%B6rling_problem&oldid=23196

External image galleries

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

• Minimal surfaces
• Differential geometry