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Characterization of continuity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds

Hyunsoo Ahn

Abstract

For a compact subset in a compact Hermitian manifold, we prove that the continuity of the extremal function at a given point in the set is a local property and that the continuity of a weighted extremal function follows from the continuities of the extremal function and the weight function. These results are generalizations of the results of Nguyen \cite{Ng24} on compact Kähler manifolds. Moreover, for a compact subset in a compact Hermitian manifold, we characterize the continuity of the extremal function via the local \(L\)-regularity, which is equivalent to the weak local \(L\)-regularity.

Characterization of continuity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds

Abstract

For a compact subset in a compact Hermitian manifold, we prove that the continuity of the extremal function at a given point in the set is a local property and that the continuity of a weighted extremal function follows from the continuities of the extremal function and the weight function. These results are generalizations of the results of Nguyen \cite{Ng24} on compact Kähler manifolds. Moreover, for a compact subset in a compact Hermitian manifold, we characterize the continuity of the extremal function via the local -regularity, which is equivalent to the weak local -regularity.

Paper Structure

This paper contains 5 sections, 15 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Continuity of extremal functions
  3. Basic properties of extremal functions on a compact Hermitian manifold
  4. Continuity of extremal functions on a compact Hermitian manifold
  5. Appendix

Key Result

Theorem 1.4

Let $K$ be a compact subset of a compact Hermitian manifold $X$ and $a\in K$. Let $\bar{B}(a,r)$ be a closed holomorphic coordinate ball in $X$ with the center $a\in K$ and the finite radius $r>0$. Then the following items hold: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2: local $L$-regularity in a complex manifold
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 25 more