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Using the Schwarzian derivative we construct a sequence \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) of meromorphic differentials on every non-flat oriented minimal surface in Euclidean \(3\)-space. The differentials \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) 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.

Related project(s):
26Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)