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Local and global plurisubharmonic defining functions

Local and global plurisubharmonic defining functions

Let D be a bounded pseudoconvex domain with real-analytic boundary in C 2 which satisfies a geometric condition on the set of points where the Levi form degenerates.If locally D has smooth defining functions plurisubharmonic on the boundary, then D has a global smooth defining function plurisubharmonic on the boundary.