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 $det(J_f)geq0$, i.e. $f$ is orientation-preserving. We give a complete characterization of the fibers of $f$ making use of a new notion we intro-duce 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.
On a class of orientation-preserving maps of R^4 / Perotti, Alessandro; Ghiloni, Riccardo. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - STAMPA. - 2021/31:3(2021), pp. 2383-2415. [10.1007/s12220-020-00356-8]
On a class of orientation-preserving maps of R^4
Perotti, Alessandro;Ghiloni, Riccardo
2021-01-01
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 $det(J_f)geq0$, i.e. $f$ is orientation-preserving. We give a complete characterization of the fibers of $f$ making use of a new notion we intro-duce 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.File | Dimensione | Formato | |
---|---|---|---|
Orientation-preserved-JGA R1.pdf
Solo gestori archivio
Tipologia:
Pre-print non referato (Non-refereed preprint)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
517.31 kB
Formato
Adobe PDF
|
517.31 kB | Adobe PDF | Visualizza/Apri |
JGEA OnAClassOfOrientation-PreservingMapsOfR4.pdf
Solo gestori archivio
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
527.39 kB
Formato
Adobe PDF
|
527.39 kB | Adobe PDF | Visualizza/Apri |
JGEA OnAClassOfOrientation-PreservingMapsOfR4_printed.pdf
Solo gestori archivio
Descrizione: Author's personal copy
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
568.33 kB
Formato
Adobe PDF
|
568.33 kB | Adobe PDF | Visualizza/Apri |
Orientation-preserved-arXiv v2.pdf
accesso aperto
Descrizione: ArXiv
Tipologia:
Pre-print non referato (Non-refereed preprint)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
389.41 kB
Formato
Adobe PDF
|
389.41 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione