Failure of averaging on multiply connected domains
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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     author = {Barrett, David E.},
     title = {Failure of averaging on multiply connected domains},
     journal = {Annales de l'Institut Fourier},
     pages = {357--370},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {2},
     year = {1990},
     doi = {10.5802/aif.1216},
     zbl = {0726.30034},
     mrnumber = {92b:30050},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1216/}
}
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Barrett, David E. Failure of averaging on multiply connected domains. Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 357-370. doi : 10.5802/aif.1216. https://aif.centre-mersenne.org/articles/10.5802/aif.1216/

[AAK] V. M. Adamyan, D. Z. Arov and M. G. Krein, Infinite Hankel matrices and generalized problems of Caratheodory, Fejer and I. Schur, Functional Anal. Appl., 2 (1986), 269-281. | Zbl

[AW] H. Alexander and J. Wermer, Polynomial hulls with convex fibers, Math. Ann., 271 (1985), 99-109. | MR | Zbl

[B] B. Berndtsson, Leviflat surfaces with circular sections, Proceedings of the Mittag-Leffler Institute (1987-1988), Princeton University Press Mathematical Notes, 38 (1990). | Zbl

[BR] B. Berndtsson and T. Ransford, Analytic multifunctions, the ð-equation and a proof of the corona theorem, Pac. J. Math., 124 (1986), 57-72. | MR | Zbl

[EM1] C. J. Earle and J. Marden, Projections to automorphic functions, Proc. Amer. Math. Soc., 19 (1968), 274-278. | MR | Zbl

[EM2] C. J. Earle and J. Marden, On Poincaré series with applications to Hp spaces on bordered Riemann surfaces, Illinois J. Math., 13 (1969), 202-219. | Zbl

[FR] F. Forelli, Bounded holomorphic functions and projections, Illinois J. Math., 10 (1966), 367-380. | MR | Zbl

[Fn1] F. Forstnerĭc, Polynomially convex hulls with piecewise smooth boundaries, Math. Ann., 276 (1986), 97-104. | MR | Zbl

[Fn2] F. Forstneriĭc, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. Jour., 37 (1988), 869-889. | MR | Zbl

[G] J. Garnett, Bounded analytic functions, Academic Press, London and New York, 1981. | MR | Zbl

[Hu] H. Huber, Uber analytische Abbildungen von Ringgebieten in Ringebiete, Compos. Math., 9 (1951), 161-168. | Numdam | MR | Zbl

[JM] P. Jones and D. Marshall, Critical points of Green's function, harmonic measure, and the corona problem, Ark. Mat., 23 (1985), 281-314. | MR | Zbl

[K] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970. | MR | Zbl

[N] R. Nevanlinna, Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn., 32 (1929). | JFM

[Na] R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam, 1968. | MR | Zbl

[Sch] S. Scheinberg, thesis, Princeton University, 1962.

[Sk1] S. Slodkowski, An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra, Trans. Amer. Math. Soc., 294 (1986), 367-377. | MR | Zbl

[Sk2] S. Slodkowski, On bounded analytic functions in finitely connected domains, Trans. Amer. Math. Soc., 300 (1987), 721-736. | MR | Zbl

[St] E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math. Soc., 120 (1965), 255-285. | MR | Zbl

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