On convex to pseudoconvex mappings
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- by S. Ivashkovich PDF
- Proc. Amer. Math. Soc. 138 (2010), 899-906 Request permission
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.
References
- M. G. Darboux, Sur le théorème fondamental de la géométrie projective, Math. Ann. 17 (1880), no. 1, 55–61 (French). MR 1510050, DOI 10.1007/BF01444119
- P. Schöpf, Konvexitätstreue und Linearität von Abbildungen, Math. Z. 177, no. 4 (1981), 533-540.
- J. L. Walsh, On the transformation of convex point sets, Ann. of Math. (2) 22 (1921), no. 4, 262–266. MR 1502587, DOI 10.2307/1967907
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
- Received by editor(s): March 10, 2009
- Published electronically: November 5, 2009
- Communicated by: Franc Forstneric
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 899-906
- MSC (2010): Primary 32F10; Secondary 52A20, 32U15
- DOI: https://doi.org/10.1090/S0002-9939-09-10200-9
- MathSciNet review: 2566556