Failure of averaging on multiply connected domains

Barrett, David E.

Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 357-370.

Nous démontrons que pour toute surface de Riemann ouverte $X$, de groupe fondamental non-abélien, il existe une fonction analytique multiforme $f$ sur $X$ telle que l’enveloppe convexe par fibre du graphe de $f$ ne contienne le graphe d’aucune fonction holomorphe uniforme sur $X$.

We show that for every open Riemann surface $X$ with non-abelian fundamental group there is a multiple-valued function $f$ on $X$ such that the fiberwise convex hull of the graph of $f$ fails to contain the graph of a single-valued holomorphic function on $X$.

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