Plateau’s problem asks to find a minimal surface that spans a given closed curve. This is a global question, and the answers are delicate. In contrast, there is a much simpler local problem, posed by Emanuel Gabriel Björling in 1844:
For a given space curve and normal field along the curve, find a minimal surfaces that contains the curve and has the given normal as surface normal. For instance, for a circle with outer normal we expect a catenoid, and for a straight line with a normal that rotates with constant speed a helicoid.
In his 1844 paper, Björling proved that for any given real analytic space curve and real analytic normal field along the curve, there is a unique local solution. Later, Hermann Amandus Schwarz gave an explicit formula for the Weierstrass data of the solution.
The fact that the solution of Björling’s problem is rather simple has been quite fertile. For instance, one can let a normal rotate about a circle and get Pablo Mira’s circular helicoids. However, one quickly finds oneself in the default situation of the wizard’s apprentice who has learned how to use spell but not acquired knowledge about the consequences.
For instance, who would have thought that if one starts with a planar cycloid as a curve and takes as normal the normal vector to the curve, the resulting surface (already known to Eugène Charles Catalan) contains parabolas as geodesics? Also, the formulas that Schwarz gave us are not always easy to integrate. The helicoids winding along logarithmic helices below (due to Christine Breiner and Stephen Kleene) can be explicitly described, but the formulas span an entire page.
So it would be desirable to have some control about the global nature of the solutions, and also some insight when to expect explicit formulas. We will earn about this next week.
E.-G. Björling: In integrationem aequationis Derivatarum partialium superficiei, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt signoque contrario, Archiv der Mathematik IV (1844), 290-315
Mathematica Notebook with Examples
Björling Surfaces Repository Page