A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space

TL;DR

This paper establishes a uniqueness theorem for the Gauss maps of complete minimal surfaces in four-dimensional Euclidean space, extending classical value-sharing results for meromorphic functions.

Contribution

It provides a new unicity theorem for Gauss maps of minimal surfaces in 4D, generalizing Nevanlinna's classical result to higher-dimensional geometric contexts.

Findings

Gauss maps sharing five distinct values are identical

Extension of Nevanlinna's theorem to minimal surfaces in 4D

New criteria for the uniqueness of minimal surface Gauss maps

Abstract

The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.