Björling problem

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.

is a real analytic curve in

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

[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.