The Schwarzian derivative and the degree of a classical minimal surface

TL;DR

This paper introduces a sequence of invariant meromorphic differentials derived from the Schwarzian derivative on minimal surfaces, defining a degree concept for these surfaces and exploring their approximation properties.

Contribution

It constructs a new sequence of invariants for minimal surfaces based on the Schwarzian derivative and defines the degree of a minimal surface, revealing new classification and approximation results.

Findings

Well-known minimal surfaces have small degree, including Enneper's surface and the helicoid.

Minimal surfaces can be approximated by surfaces of increasing degree.

The differentials depend only on the induced metric and its derivatives.

Abstract

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.