A factorization of a super-conformal map

TL;DR

This paper introduces a factorization method for super-conformal maps and minimal surfaces using quaternionic modeling, revealing properties analogous to classical complex analysis theorems.

Contribution

It provides a novel factorization approach for super-conformal maps and minimal surfaces, extending classical complex analysis results to quaternionic and four-dimensional contexts.

Findings

Factorization of super-conformal maps into holomorphic or meromorphic components.

Analogues of classical theorems like Liouville, Schwarz, and Weierstrass are established.

Relation between zeros of minimal surfaces and branch points of super-conformal maps.

Abstract

A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a meromorphic function. Analogs of the Liouville theorem, the Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem, the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and branch points of a super-conformal map are obtained.