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.

KEYWORDS:

• Caffarelli–Nirenberg–Spruck model
• harmonic morphisms
• m-subharmonic functions
• holomorphic maps
• plurisubharmonic functions
• subharmonic functions

AMS SUBJECT CLASSIFICATIONS:

• Primary: 58E20
• 32A10
• 32U05
• Secondary: 31C45
• 15A18

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second-named author was partially supported by Narodowe Centrum Nauki (NCN) grant DEC-2013/08/A/ST1/00312.