Core Sets in Kahler Manifolds

TL;DR

This paper explores the structure of core sets in Kahler manifolds using m-subharmonic functions, establishing the relationship between m-harmonic and pluriharmonic functions and introducing the concept of m-core based on m-pseudoconcavity.

Contribution

It introduces the concept of m-core in Kahler manifolds and characterizes m-harmonic functions as pluriharmonic in higher dimensions.

Findings

m-harmonic functions are pluriharmonic in complex dimension at least 2

m-core sets are defined via m-pseudoconcavity

structure of core sets studied in non-compact Kahler manifolds

Abstract

The primary objective of this paper is to study core sets in the setting of m-subharmonic functions on the class of (non-compact) Kahler manifolds. Core sets are investigated in different aspects by considering various classes of plurisubharmonic functions. One of the crucial concepts in studying the structure of this kind of sets is the pseudoconcavity. In a more general way, we will have the structure of core defined with respect to the m-subharmonic functions, which we call m-core in our setting, in terms of m-pseudoconcave sets. In the context of m-subharmonic functions, we define m-harmonic functions and show that, in $\mathbb{C}^{n},\,(n \geq 2)$ and more generally in any Kahler manifold of dimension at least 2, m-harmonic functions are pluriharmonic functions for $m \geq 2$.