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Core Sets in Kahler Manifolds

Nihat Gökhan Gögüş, Ozan Günyüz, Özcan Yazıcı

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 $m \geq 2$.

Core Sets in 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 and more generally in any Kahler manifold of dimension at least 2, m-harmonic functions are pluriharmonic functions for .
Paper Structure (3 sections, 18 theorems, 37 equations)

This paper contains 3 sections, 18 theorems, 37 equations.

Table of Contents

  1. INTRODUCTION, PRELIMINARIES AND THE RESULT
  2. STRUCTURE OF THE $m$-CORE
  3. a characterization of pluriharmonic functions

Key Result

Theorem 1.1

The core $\mathbf{c}_{m}(M)$ can be disintegrated into $m$-pseudoconcave subsets such that every upper bounded continuous $m$-subharmonic function on $M$ is constant on each of these sets.

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 22 more