2023/01/27 by Thomas Mettler, Mettler, Thomas, Lukas Poerschke +1
Mathematics · #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2301.11700
openalex publication_date 2023/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using the Schwarzian derivative we construct a sequence (Pℓ)ℓ \geqslant 2 of meromorphic differentials on every non-flat oriented minimal surface in Euclidean 3-space. The differentials (Pℓ)ℓ \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.