Every biregular function is a biholomorphic map
The set of (left) Fueter-biregular functions of one quaternionic variable is considered. The aim of the paper is to give a characterization of biregular functions by means of holomorphy with respect to the orthogonal almost complex structures defined on dense, open subsets of a set \(\Omega\) in \({\mathbb H}\). A criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes (pseudo) holomorphic functions among regular functions is proved. A formula is presented which expresses the determinant of the energy quadric as a polynomial in the first derivatives of the function. This formula is used to show that, for every biregular function, there exist a dense, open subset \(\Omega^{\prime}\) of \(\Omega\) and an almost complex \(p(z)\) on \(\Omega^{\prime}\) such that the function belongs to the set \(\text{Hol}_p(\Omega^{\prime}, {\mathbb H})\).
scientific article; zbMATH DE number 5593792