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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On convex to pseudoconvex mappings


Author: S. Ivashkovich
Journal: Proc. Amer. Math. Soc. 138 (2010), 899-906
MSC (2010): Primary 32F10; Secondary 52A20, 32U15
Published electronically: November 5, 2009
MathSciNet review: 2566556
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Abstract: In the works of Darboux and Walsh, it was remarked that a one-to-one self-mapping of $ \mathbb{R}^{3}$ which sends convex sets to convex ones is affine. It can be remarked also that a $ \mathcal{C}^{2}$-diffeomorphism $ F:U\to U^{'}$ between two domains in $ \mathbb{C}^{n}$, $ n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic.

In this paper we are interested in the self-mappings of $ \mathbb{C}^{n}$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: A $ \mathcal{C}^{2}$- diffeomorphism $ F:U'\to U$ (where $ U', U\subset \mathbb{C}^{n}$ are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map $ \Phi := F^{-1}$ is weakly pluriharmonic, i.e., if it satisfies some nice second order PDE very close to $ \partial \bar{\partial}\Phi = 0$. In fact all pluriharmonic $ \Phi$'s do satisfy this equation, but there are also other solutions.


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Additional Information

S. Ivashkovich
Affiliation: UFR de Mathématiques, Université de Lille-1, 59655 Villeneuve d’Ascq, France – and – Institute of Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Naukova 3B, 79601 Ukraine
Email: ivachkov@math.univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10200-9
PII: S 0002-9939(09)10200-9
Keywords: Convex, pseudoconvex, pluriharmonic
Received by editor(s): March 10, 2009
Published electronically: November 5, 2009
Communicated by: Franc Forstneric
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.