On convex to pseudoconvex mappings TeX Export

@article{citeulike:4166169, abstract = {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 \$\Phi\deff F^{-1}\$ is weakly pluriharmonic, i.e. it satisfies some nice second order PDE very close to \$\d\bar\d \Phi = 0\$.} In fact all pluriharmonic \$\Phi\$-s do satisfy this equation, but there are also other solutions.}, archivePrefix = {arXiv}, author = {Ivashkovich, S.}, citeulike-article-id = {4166169}, citeulike-linkout-0 = {http://arxiv.org/abs/0903.1787}, citeulike-linkout-1 = {http://arxiv.org/pdf/0903.1787}, day = {10}, eprint = {0903.1787}, keywords = {3d, convex, geometry}, month = {Mar}, posted-at = {2009-03-11 20:38:32}, priority = {2}, title = {On convex to pseudoconvex mappings}, url = {http://arxiv.org/abs/0903.1787}, year = {2009} }

TY - JOUR ID - citeulike:4166169 L3 - citeulike-article-id:4166169 N2 - 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 $\Phi\deff F^{-1}$ is weakly pluriharmonic, i.e. it satisfies some nice second order PDE very close to $\d\bar\d \Phi = 0$.} In fact all pluriharmonic $\Phi$-s do satisfy this equation, but there are also other solutions. TI - On convex to pseudoconvex mappings KW - 3d KW - convex KW - geometry AU - Ivashkovich, S PY - 2009/Mar/10/ UR - http://arxiv.org/abs/0903.1787 ER -