Some remarks on Hartogs’ extension lemma

Černe, Miran; Flores, Manuel

doi:10.1090/S0002-9939-10-10367-0

Some remarks on Hartogs’ extension lemma
HTML articles powered by AMS MathViewer

by Miran Černe and Manuel Flores PDF

Proc. Amer. Math. Soc. 138 (2010), 3603-3608 Request permission

Abstract:

Motivated by a result and a question by E. M. Chirka we consider the Hartogs’ extension property for some connected sets in $\mathbb {C}^ 2$ of the form $K=\Sigma \cup (\partial \Delta \times \overline {\Delta })$, where $\Sigma$ is a possibly nonconnected compact subset of $\overline {\Delta }\times \overline {\Delta }\subset \mathbb {C}^ 2$.

References

David E. Barrett and Gautam Bharali, The role of Fourier modes in extension theorems of Hartogs-Chirka type, Math. Z. 249 (2005), no. 4, 883–901. MR 2126221, DOI 10.1007/s00209-004-0742-0
Gautam Bharali, Some generalizations of Chirka’s extension theorem, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3665–3669. MR 1860501, DOI 10.1090/S0002-9939-01-06020-8
Miran Černe and Manuel Flores, Quasilinear $\overline \partial$-equation on bordered Riemann surfaces, Math. Ann. 335 (2006), no. 2, 379–403. MR 2221118, DOI 10.1007/s00208-006-0755-6
E. M. Chirka, The generalized Hartogs lemma and the nonlinear $\overline \partial$-equation, Complex analysis in modern mathematics (Russian), FAZIS, Moscow, 2001, pp. 19–31 (Russian, with Russian summary). MR 1833502
Evgeni Chirka and Jean-Pierre Rosay, Remarks on the proof of a generalized Hartogs lemma, Ann. Polon. Math. 70 (1998), 43–47. Complex analysis and applications (Warsaw, 1997). MR 1668715, DOI 10.4064/ap-70-1-43-47
Evgeni M. Chirka and Edgar Lee Stout, A Kontinuitätssatz, Topics in complex analysis (Warsaw, 1992) Banach Center Publ., vol. 31, Polish Acad. Sci. Inst. Math., Warsaw, 1995, pp. 143–150. MR 1341384
Franc Forstnerič, Stein domains in complex surfaces, J. Geom. Anal. 13 (2003), no. 1, 77–94. MR 1967038, DOI 10.1007/BF02930998
S. Ivashkovich and V. Shevchishin, Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls, Invent. Math. 136 (1999), no. 3, 571–602. MR 1695206, DOI 10.1007/s002220050319
S. Yu. Nemirovskiĭ, Complex analysis and differential topology on complex surfaces, Uspekhi Mat. Nauk 54 (1999), no. 4(328), 47–74 (Russian, with Russian summary); English transl., Russian Math. Surveys 54 (1999), no. 4, 729–752. MR 1741278, DOI 10.1070/rm1999v054n04ABEH000179
R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5
Jean Pierre Rosay, A counterexample related to Hartogs’ phenomenon (a question by E. Chirka), Michigan Math. J. 45 (1998), no. 3, 529–535. MR 1653267, DOI 10.1307/mmj/1030132298