The classification of holomorphic (m, n)-subharmonic morphisms

Abstract

We study the problem of classifying the holomorphic (m, n)-subharmonic morphisms in complex space. This determines which holomorphic mappings preserves m-subharmonicity in the sense that the composition of the holomorphic mapping with a m-subharmonic functions is n-subharmonic. We show that there are three different scenarios depending on the underlying dimensions, and the model itself. Either the holomorphic mappings are just the constant functions, or up to composition with a homotethetic map, canonical orthogonal projections. Finally, there is a more intriguing case when subharmonicity is gained in the sense of the Caffarelli-Nirenberg-Spruck framework.2010 Mathematics Subject Classification. Primary 58E20, 32A10, 32U05; Secondary 31C45, 15A18.