Twistor lifts and factorization for conformal maps from a surface to the Euclidean four-space

TL;DR

This paper explores the use of twistor lifts to analyze conformal maps from Riemann surfaces to Euclidean four-space, introducing a local factorization of differentials and applying it to bound areas of super-conformal maps.

Contribution

It introduces a novel factorization method for differentials of conformal maps via twistor lifts, providing new insights into their geometric properties.

Findings

Factorization of differentials of conformal maps achieved

Upper bound for the area of super-conformal maps near branch points established

Enhanced understanding of conformal maps through twistor theory

Abstract

A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point.