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Every Biregular Function Is a Biholomorphic Map

Authors:

Alessandro Perotti

Institution:

(1) Department of Mathematics, University of Trento, Via Sommarive, 14, I-38050 Povo Trento, Italy

Abstract:

We study Fueter-biregular functions of one quaternionic variable. We consider left-regular functions in the kernel of the Cauchy–Riemann operator

${\mathcal{D}} = \frac{\partial f}{\partial x_{0}} + i\frac{\partial f}{\partial x_{1}} + j\frac{\partial f}{\partial x_2} - k\frac{\partial f}{\partial x_3} = 2\left(\frac{\partial}{\partial \bar{z}_{1}} + j\frac{\partial}{\partial\bar{z}_2}\right)$

. A quaternionic function $f \in C^{1}(\Omega)$ is biregular if ${\mathcal{D}}(f) = 0$ on Ω, f is invertible and ${\mathcal{D}}(f^{-1}) = 0$ . Every continuous map p from Ω to the sphere ${\mathbb{S}}^{2}$ of unit imaginary quaternions induces an almost complex structure J_p on the tangent bundle of ${\mathbb{H}}$ . Let $H ol_{p}(\Omega, {\mathbb{H}})$ be the space of (pseudo)holomorphic maps from (Ω, J_p) to ( ${\mathbb{H}}, L_{p}$ ), where L_p is the almost complex structure defined by left multiplication by p. Every element of $H ol_{p}(\Omega, {\mathbb{H}})$ is regular, but there exist regular functions that are not holomorphic for any p. The space of biregular functions contains the invertible elements of the spaces $H ol_{p}(\Omega, {\mathbb{H}})$ . By means of a criterion, based on the energy-minimizing property of holomorphic maps, that characterizes holomorphic functions among regular functions, we show that every biregular function belongs to some space $H ol_{p}(\Omega, {\mathbb{H}})$ . Received: October, 2007. Accepted: February, 2008.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 32A30 Secondary 30G35

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