The Björling problem is in its nature a Cauchy problem, and its solution enjoys benefits and side effects of such problems: We have, for an analytic curve and normal, the uniqueness and existence of a minimal surface with these Cauchy data.

Things are even better than in general, because the solution can be computed as an integral. The side effect is that we have little flexibility, and little knowledge about the global appearance of the solution far away from the curve.

It turns out that one can do something about it, if one is willing to sacrifice the precise shape of the curve. Instead of prescribing the complete space curve (x(t), y(t), z(t)), we might be content prescribing just the planar projection (x(t), y(t)). Then there is a way to explicitly determine a third coordinate z(t) so that one can obtain more global control over the Björling surfaces, like finite total curvature, completeness, and flexible speed of a rotating normal.

The construction also comes with a parameter that allows to keep a planar curve closed after lifting it into space.

This raises the intriguing question whether there are similar phenomena for other Cauchy problems: Does forgetting one dimension of the Cauchy data allow for some global control?

Resources

Repository Overview: Björling Surfaces

R. López, M. Weber: Explicit Björling Surfaces with Prescribed Geometry

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