The Schwarzian Derivative and the Degree of a Classical Minimal Surface

Using the Schwarzian derivative we construct a sequence (Pl)l⩾2 of meromorphic differentials on every non-flat oriented minimal surface in Euclidean 3-space. The differentials (Pl)l⩾2 are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree n if its n-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper's surface, the helicoid/catenoid and the Scherk—as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.