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On a Class of Orientation-Preserving Maps of \(\pmb {\mathbb {R}}^4\)

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Abstract

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian matrix \(J_f\) of a slice regular function f proving in particular that \(\det (J_f)\ge 0\), i.e., f is orientation-preserving. We give a complete characterization of the fibers of f making use of a new notion we introduce here, the one of wing of f. We investigate the singular set \(N_f\) of f, i.e., the set in which \(J_f\) is singular. The singular set \(N_f\) turns out to be equal to the branch set of f, i.e., the set of points y such that f is not a homeomorphism locally at y. We establish the quasi-openness properties of f. As a consequence we deduce the validity of the Maximum Modulus Principle for f in its full generality. Our results are sharp as we show by explicit examples.

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Acknowledgements

This work was supported by GNSAGA of INdAM, and by the grants “Progetto di Ricerca INdAM, Teoria delle funzioni ipercomplesse e applicazioni”, and PRIN “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” of the Italian Ministry of Education.

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  1. Department of Mathematics, University of Trento, 38123, Povo-Trento, Italy

    Riccardo Ghiloni & Alessandro Perotti

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  1. Riccardo Ghiloni
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  2. Alessandro Perotti
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Correspondence to Alessandro Perotti.

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Ghiloni, R., Perotti, A. On a Class of Orientation-Preserving Maps of \(\pmb {\mathbb {R}}^4\). J Geom Anal 31, 2383–2415 (2021). https://doi.org/10.1007/s12220-020-00356-8

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