The Schwarzian Derivative and the Degree of a Classical Minimal Surface

Abstract

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

Keywords:

• minimal surfaces
• Schwarzian derivative
• meromorphic differentials

Acknowledgments

This article is based on the doctoral thesis of L.P. The authors were partially supported by the priority programme SPP 2026 “Geometry at Infinity” of DFG. The authors are grateful to Jacob Bernstein for several helpful discussions regarding the content of this article. The authors also express their gratitude to the anonymous referees for their reports, which helped to enhance the clarity of this paper.