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Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 11 (2016), 33 -- 75

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This work is licensed under a Creative Commons Attribution 4.0 International License.

THE LEVI PROBLEM IN ℂn: A SURVEY

Harry J. Slatyer

Abstract. We discuss domains of holomorphy and several notions of pseudoconvexity (drawing parallels with the corresponding concepts from geometric convexity), and present a mostly self-contained solution to the Levi problem. We restrict our attention to domains of ℂn.

2010 Mathematics Subject Classification: 32E40; 32-01
Keywords: classical Levi problem, survey

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References

  1. H.~Behnke and K.~Stein. Konvergente Folgen nichtschlichter Regularitätsbereiche. Annali di Matematica Pura ed Applicata, 28(1):317--326, 1949. MR35338(11,719c). Zbl 0038.05402.

  2. L.~Bers. Introduction to Several Complex Variables: Lectures Delivered at the Courant Institute of Mathematical Sciences, New York University, 1962-1963. Courant Institute of Mathematical Sciences, New York University, 1964.

  3. H.~Boas. Lecture notes on several complex variables. rlhttp://www.math.tamu.edu/~boas/courses/650-2007c/notes.pdf, accessed August 2012.

  4. S.~Bochner. Analytic and meromorphic continuation by means of Green's formula. Annals of Mathematics, 44(4):652--673, 1943. MR9206(5,116f). Zbl 0060.24206.

  5. H.~J. Bremermann. Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum von n komplexen Veränderlichen. Mathematische Annalen, 128:63--91, 1954. MR71088(17,82a). Zbl 0056.07801.

  6. H.~Cartan and P.~Thullen. Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen. Mathematische Annalen, 106:617--647, 1932. MR1512777. Zbl 0004.35704.

  7. H.~Grauert. On Levi's problem and the imbedding of real-analytic manifolds. Annals of Mathematics, 68(2):460--472, 1958. MR0098847(20 #5299). Zbl 0108.07804.

  8. R.~Gunning and H.~Rossi. Analytic Functions of Several Complex Variables. Reprint of the 1965 original. AMS Chelsea Publishing, Providence, 2009. MR180696(31 #4927). Zbl 1204.01045.

  9. F.~Hartogs. Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen. Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse, 36:223--242, 1906. JFM 37.0443.01.

  10. L.~Hörmander. L2 estimates and existence theorems for the øverline∂ operator. Acta Mathematica, 113:89--152, 1965. MR179443(31 #3691). Zbl 0158.11002.

  11. L.~Hörmander. An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990. MR1045639(91a:32001). Zbl 0685.32001.

  12. S.~Krantz. Function Theory of Several Complex Variables. Reprint of the 1992 edition. AMS Chelsea Publishing, Providence, 2001. MR1846625(2002e:32001). Zbl 1087.32001.

  13. P.~Lelong. Les fonctions plurisousharmoniques. Annales scientifiques de l'École Normale Supérieure, 62:301--338, 1945. MR18304(8,271f). Zbl 0061.23205.

  14. E.~Levi. Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse. Annali di Matematica Pura ed Applicata, 17:61--87, 1910. JFM 41.0487.01.

  15. E.~Levi. Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse. Annali di Matematica Pura ed Applicata, 18:69--79, 1911. JFM 42.0449.02.

  16. E.~Martinelli. Alcuni teoremi integrali per le funzioni analitiche di piu variabili complesse. Memorie della Reale Accademia d'Italia, Classe di Scienze Fisiche, Matematiche e Naturali, 7. Serie, 9:269--283, 1938. Zbl 0022.24002.

  17. A.~P. Morse. The behavior of a function on its critical set. Annals of Mathematics, 40(2):62--70, 1939. MR1503449. Zbl 0020.01205.

  18. R.~Narasimhan. The Levi problem for complex spaces. Mathematische Annalen, 142:355--365, 1961. MR148943(26 #6439). Zbl 0106.28603.

  19. F.~Norguet. Sur les domaines d'holomorphie des fonctions uniformes de plusieurs variables complexes. (Passage du local au global.). Bulletin de la Société Mathématique de France, 82:137--159, 1954. MR71087(17,81c). Zbl 0056.07701.

  20. K.~Oka. Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes. Tôhoku Mathematical Journal, 49:15--52, 1942. MR14470(7,290a). Zbl 0060.24006.

  21. K.~Oka. Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur. Japanese Journal of Mathematics, 23:97--155, 1953. MR71089(17,82b). Zbl 0053.24302.

  22. R.~M. Range. Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics. Springer, New York, 1986. MR847923(87i:32001). Zbl 0591.32002.

  23. B.~V. Shabat. Introduction to Complex Analysis. Part II. Functions of Several Variables. Translations of Mathematical Monographs, 110. American Mathematical Society, Providence, 1992. MR1192135(93g:32001). Zbl 0799.32001.

  24. Y.-T. Siu. Pseudoconvexity and the problem of Levi. Bulletin of the American Mathematical Society, 84(4):481--512, 1978. MR477104(57 #16648). Zbl 0423.32008.

  25. H.~J. Slatyer. The Levi Problem: references. rlhttp://levi-problem.appspot.com/references.html, accessed April 2016.

  26. V.~S. Vladimirov. Methods of the Theory of Functions of Many Complex Variables. The M.I.T. Press, Cambridge-London, 1966. MR201669(34 #1551). Zbl 0125.31904.



Harry J. Slatyer
Department of Quantum Science, The Australian National University,
Canberra, ACT 0200, Australia.
e-mail: harry.slatyer@anu.edu.au

http://www.utgjiu.ro/math/sma