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 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.