On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions (Q268419)

In this paper the authors prove results that were partially announced in their paper [Russ. Math. Surv. 69, No. 1, 159--161 (2014; Zbl 1290.31002); translation from Usp. Mat. Nauk 69, No. 1, 169--170 (2014)].NEWLINENEWLINEThe main theorems are formulated asNEWLINENEWLINETheorem 1. Let \(E=\cup_{j=1}^m\), \(E_j\subset{\mathcal E}_m\), \(f\in{\mathcal A}(E),\mu\in M(E),\mu(E_j)>0\;(1\leq j\leq m)\). Suppose thet the \(\mu\)-minimizing element in \({\mathcal K}_{E,f}\) consists of a finite number of continua. Then \(F\) is symmetric in the external field \({\mathcal V}^{-\mu}\) and \(\mathbb C\setminus F=\cup_{j=1}^m\,D_j\), where \(E_j\subset D_j\).NEWLINENEWLINEMoreover, the domains \(D_j\) and \(D_k\;(1\leq j,k\leq m)\) either do not intersect one another or coincide.NEWLINENEWLINETheorem 2. Let \(E\subset [a,b]\subset\mathbb C\) be a closed interval,\ \(\mu\in{\tilde M}(E)\). There exists a function \(f\in{\mathcal A}(E)\) such that the \(\mu\)-minimization problem NEWLINE\[NEWLINE\mathrm{cap}_{\mu}\,F=\inf_{K\in {\mathcal K}_{E,f}}\,\mathrm{cap}_{\mu}\,K, NEWLINE\]NEWLINE in the family of compacta \({\mathcal K}_{E,f} \) is not solvable.NEWLINENEWLINE\({\mathcal E}_m\) is the class of compacta \(E\subset\overline{\mathbb C}\) of the form \(E=\cup_{j=1}^m\,E_j\), where \(E_1,\dots,E_m\) are pairwise disjoint continua in \(\overline{\mathbb C}\) (some may consist of a single point),NEWLINENEWLINE\({\mathcal A}(E)\) is the class of functions defined on \(E=\cup_{j=1}^m\,E_j\), such that each restriction \(f_j=f\mid_{E_j}\,(1\leq j\leq m)\) is holomorphic on \(E_j\) and admits a continuation along every path in \(\mathbb C\), not passing through a finite pointset \(A_{f_j}\), where \(A_{f_j}\) contains at least one branch point of \(f_j\)\dotsNEWLINENEWLINEA very nicely written paper, giving ample background of historical developments and showing in a clear and concise way all the intricate steps in the proofs.