Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Relative extremal functions and characterization of pluripolar sets in complex manifolds


Authors: Armen Edigarian and Ragnar Sigurdsson
Journal: Trans. Amer. Math. Soc. 362 (2010), 5321-5331
MSC (2010): Primary 32U15; Secondary 32U20, 32E20
DOI: https://doi.org/10.1090/S0002-9947-2010-05080-9
Published electronically: May 20, 2010
MathSciNet review: 2657681
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a disc formula for the relative extremal function for a subset of a complex manifold and apply it to give a description of pluripolar sets and polynomial hulls.


References [Enhancements On Off] (What's this?)

  • 1. H. ALEXANDER AND J. WERMER, Several complex variables and Banach Algebras, 2nd ed., Springer-Verlag, 1998. MR 1482798 (98g:32002)
  • 2. E. BEDFORD, The operator $ (dd^c)^n$ on complex spaces, Lecture Notes in Math. 919, 294-323 (1982). MR 658889 (83i:32025)
  • 3. E. BEDFORD AND B. A. TAYLOR, A new capacity for plurisubharmonic functions, Acta Math. 149, 1-40 (1982). MR 674165 (84d:32024)
  • 4. S. BU AND W. SCHACHERMAYER, Approximation of Jensen measures by image meaures under holomorphic functions and applications, Trans. AMS 331, no. 2, 585-608 (1992). MR 1035999 (92h:46063)
  • 5. A. EDIGARIAN, Analytic discs method in complex analysis, Dissertationes Math. (Rozprawy Mat.) 402, 56 pp. (2002). MR 1897580 (2002m:32049)
  • 6. A. EDIGARIAN, A note on J. P. Rosay's paper, Ann. Polon. Math. 80, 125-132 (2003). MR 1972839 (2004f:32043)
  • 7. H. GRAUERT, Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z. 81, 377-391 (1963). MR 0168798 (29:6054)
  • 8. B. JOSEFSON, On the equivalence between locally polar and globally polar for plurisubharmonic functions on $ \mathbb{C}^n$, Arkiv för Mat. 16, 109-115 (1976). MR 0590078 (58:28669)
  • 9. M. KLIMEK, Pluripotential theory. Oxford University Press, London, 1991. MR 1150978 (93h:32021)
  • 10. F. L´ARUSSON AND R. SIGURDSSON, Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501, 1-39 (1998). MR 1637837 (99e:32020)
  • 11. F. L´ARUSSON AND R. SIGURDSSON, Plurisubharmonicty of envelopes of disc functionals on manifolds, J. Reine Angew. Math. 555, 27-38 (2003). MR 1956593 (2004a:32054)
  • 12. E.A. POLETSKY, Plurisubharmonic functions as solutions of variational problems, Proc. Symp. Pure Math. 52, Part 1, 163-171 (1991). MR 1128523 (92h:32022)
  • 13. E.A. POLETSKY, Holomorphic currents, Indiana Univ. Math. J. 42, no. 1, 85-144 (1993). MR 1218708 (94c:32007)
  • 14. E.A. POLETSKY, Disk envelopes of functions II, J. Functional Anal. 163, 111-132 (1999). MR 1682839 (2001c:32017)
  • 15. J.-P. ROSAY, Poletsky theory of disks on holomorphic manifolds. Indiana Univ. Math. J. 52, no. 1, 157-169 (2003). MR 1970025 (2004a:32053)
  • 16. A. SADULLAEV, Plurisubharmonic measures and capacities on complex manifolds, Uspekhi Mat. Nauk, 36, no. 4, 53-105 (1981). English translation in Russian Math. Surveys, 36, no. 4, 61-119 (1981). MR 629683 (83c:32026)
  • 17. G. STOLZENBERG, A hull with no analytic structure, J. Math. Mech., 12, 103-111 (1963). MR 0143061 (26:627)
  • 18. E.L. STOUT, Polynomial Convexity, Birkhäuser, 2007. MR 2305474 (2008d:32012)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32U15, 32U20, 32E20

Retrieve articles in all journals with MSC (2010): 32U15, 32U20, 32E20


Additional Information

Armen Edigarian
Affiliation: Institute of Mathematics, Jagiellonian University, Lojasiewicza 6, 30-348 Kraków, Poland
Email: Armen.Edigarian@im.uj.edu.pl

Ragnar Sigurdsson
Affiliation: Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
Email: ragnar@hi.is

DOI: https://doi.org/10.1090/S0002-9947-2010-05080-9
Keywords: Analytic disc, relative extremal function, pluriregular set, pluripolar set, Josefson manifold
Received by editor(s): June 28, 2008
Published electronically: May 20, 2010
Additional Notes: This work was a part of the Research Grant No. 1 PO3A 005 28, which was supported by public means in the programme for promoting science in Poland in the years 2005-2008. The first author was a fellow of Krzyżanowski Fund at the Jagiellonian University. This work was supported by the Research Fund at the University of Iceland.
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society