On convex to pseudoconvex mappings
Abstract
Description
In the works of Darboux and Walsh it was remarked that a one to one self mapping of $\rr^3$ which sends convex sets to convex ones is affine. It can be remarked also that a $\calc^2$-diffeomorphism $F:U\to U^{'}$ between two domains in $\cc^n$, $n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic.
\smallskip In this note we are interested in the self mappings of $\cc^n$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {\it A $\calc^2$ - diffeomorphism $F:U'\to U$ (where $U', U\subset \cc^n$ are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map $Φ\deff F^{-1}$ is weakly pluriharmonic, i.e. it satisfies some nice second order PDE very close to $\d\bar\d Φ= 0$.} In fact all pluriharmonic $Φ$-s do satisfy this equation, but there are also other solutions.