Plurisubharmonicity of envelopes of disc functionals on manifolds

Abstract

We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks byholomorph ic submersions from domains of holomorphyin affine space do and it is locallybounded above and upper semicontinuous in a certain weak sense. For naturallydefined classes of disc functionals on manifolds, this result reduces a propertysomewhat stronger than having a plurisubharmonic envelope to the affine case. The proof uses a recent Stein neighbourhood construction of Rosay, who proved the plurisubharmonicityof the Poisson envelope on all manifolds. As a consequence, the Riesz envelope and the Lelong envelope are plurisubharmonic on all manifolds; for the former, we make use of new work of Edigarian. The basic theoryof the three main classes of disc functionals is therebyextended to all manifolds.