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

TL;DR

This paper studies the continuity properties of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds, establishing local characterizations and generalizing previous results from Kähler to Hermitian settings.

Contribution

It generalizes the continuity characterization of extremal functions from Kähler to Hermitian manifolds and links continuity to local L-regularity and weak local L-regularity.

Findings

Continuity of extremal functions is a local property on compact Hermitian manifolds.

Weighted extremal function continuity follows from extremal and weight function continuities.

L-regularity at star centers in ^n implies local L-regularity.

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 K\"ahler manifolds. Moreover, for a compact subset in a compact Hermitian manifold, at the point level and accordingly at the global level, we characterize the continuity of the extremal function via the local \(L\)-regularity, which is equivalent to the weak local \(L\)-regularity. We also show that the \(L\)-regularity of a compact subset in \(\mathbb{C}^n\) at a star center implies the local \(L\)-regularity. Consequently, a convex compact \(L\)-regular subset in \(\mathbb{C}^n\) is locally \(L\)-regular.