Failure of averaging on multiply connected domains

David E. Barrett

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 2, page 357-370
  • ISSN: 0373-0956

Abstract

top
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 .

How to cite

top

Barrett, David E.. "Failure of averaging on multiply connected domains." Annales de l'institut Fourier 40.2 (1990): 357-370. <http://eudml.org/doc/74880>.

@article{Barrett1990,
abstract = {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 $X$.},
author = {Barrett, David E.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {357-370},
publisher = {Association des Annales de l'Institut Fourier},
title = {Failure of averaging on multiply connected domains},
url = {http://eudml.org/doc/74880},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Barrett, David E.
TI - Failure of averaging on multiply connected domains
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 2
SP - 357
EP - 370
AB - 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 $X$.
LA - eng
UR - http://eudml.org/doc/74880
ER -

References

top
  1. [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. Zbl0179.46701
  2. [AW] H. ALEXANDER and J. WERMER, Polynomial hulls with convex fibers, Math. Ann., 271 (1985), 99-109. Zbl0538.32011MR86i:32025
  3. [B] B. BERNDTSSON, Leviflat surfaces with circular sections, Proceedings of the Mittag-Leffler Institute (1987-1988), Princeton University Press Mathematical Notes, 38 (1990). Zbl0772.32014
  4. [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. Zbl0602.32002MR87j:30082
  5. [EM1] C. J. EARLE and J. MARDEN, Projections to automorphic functions, Proc. Amer. Math. Soc., 19 (1968), 274-278. Zbl0159.11301MR37 #412
  6. [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. Zbl0169.10202
  7. [FR] F. FORELLI, Bounded holomorphic functions and projections, Illinois J. Math., 10 (1966), 367-380. Zbl0141.31401MR33 #1754
  8. [Fn1] F. FORSTNERĬC, Polynomially convex hulls with piecewise smooth boundaries, Math. Ann., 276 (1986), 97-104. Zbl0585.32016MR87m:32036
  9. [Fn2] F. FORSTNERIĬC, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. Jour., 37 (1988), 869-889. Zbl0647.32017MR90g:32018
  10. [G] J. GARNETT, Bounded analytic functions, Academic Press, London and New York, 1981. Zbl0469.30024MR83g:30037
  11. [Hu] H. HUBER, Uber analytische Abbildungen von Ringgebieten in Ringebiete, Compos. Math., 9 (1951), 161-168. Zbl0043.30201MR13,337d
  12. [JM] P. JONES and D. MARSHALL, Critical points of Green's function, harmonic measure, and the corona problem, Ark. Mat., 23 (1985), 281-314. Zbl0589.30028MR87h:30101
  13. [K] S. KOBAYASHI, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970. Zbl0207.37902MR43 #3503
  14. [N] R. NEVANLINNA, Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn., 32 (1929). JFM55.0768.03
  15. [Na] R. NARASIMHAN, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam, 1968. Zbl0188.25803MR40 #4972
  16. [Sch] S. SCHEINBERG, thesis, Princeton University, 1962. 
  17. [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. Zbl0594.32008MR87e:32021
  18. [Sk2] S. SLODKOWSKI, On bounded analytic functions in finitely connected domains, Trans. Amer. Math. Soc., 300 (1987), 721-736. Zbl0629.32003MR88e:32024
  19. [St] E. L. STOUT, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math. Soc., 120 (1965), 255-285. Zbl0154.32903MR32 #1358

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.