Björling problem
In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal. 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 is a real analytic curve in ℝ3 defined over an interval I, with and a vector field along c such that and, then the following surface is minimal:
where,, and is a simply connected domain where the interval is included and the power series expansions of and 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 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.- Björling Surfaces, at the Indiana Minimal Surface Archive: http://www.indiana.edu/~minimal/archive/Bjoerling/index.html
