The Lu Qi-Keng conjecture fails generically
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- by Harold P. Boas PDF
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Abstract:
The bounded domains of holomorphy in $\mathbb {C}^n$ whose Bergman kernel functions are zero-free form a nowhere dense subset (with respect to a variant of the Hausdorff distance) of all bounded domains of holomorphy.References
- Harold P. Boas, Counterexample to the Lu Qi-Keng conjecture, Proc. Amer. Math. Soc. 97 (1986), no. 2, 374–375. MR 835902, DOI 10.1090/S0002-9939-1986-0835902-8
- Harold P. Boas and Emil J. Straube, Equivalence of regularity for the Bergman projection and the $\overline \partial$-Neumann operator, Manuscripta Math. 67 (1990), no. 1, 25–33. MR 1037994, DOI 10.1007/BF02568420
- Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, DOI 10.1142/S0129167X90000034
- John E. Fornæss and Bill Zame, Runge exhaustions of domains in $\textbf {C}^n$, Math. Z. 194 (1987), no. 1, 1–5. MR 871214, DOI 10.1007/BF01168001
- John Erik Fornaess and Edgar Lee Stout, Spreading polydiscs on complex manifolds, Amer. J. Math. 99 (1977), no. 5, 933–960. MR 470251, DOI 10.2307/2373992
- R. E. Greene and Steven G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 179–198. MR 627758
- Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
- Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
- J. J. Kohn, Global regularity for $\bar \partial$ on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273–292. MR 344703, DOI 10.1090/S0002-9947-1973-0344703-4
- Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
- Q.-k. Lu, On Kaehler manifolds with constant curvature, Chinese Math.–Acta 8 (1966), 283–298. MR 0206990
- I. Ramadanov, Sur une propriété de la fonction de Bergman, C. R. Acad. Bulgare Sci. 20 (1967), 759–762 (French). MR 226042
- I. P. Ramadanov, Some applications of the Bergman kernel to geometrical theory of functions, Complex analysis (Warsaw, 1979) Banach Center Publ., vol. 11, PWN, Warsaw, 1983, pp. 275–286. MR 737768
- N. V. Shcherbina, Decomposition of a common boundary of two domains of holomorphy into analytic curves, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1106–1123, 1136 (Russian). MR 675533
- Maciej Skwarczyński, Biholomorphic invariants related to the Bergman function, Dissertationes Math. (Rozprawy Mat.) 173 (1980), 59. MR 575756
- Nobuyuki Suita and Akira Yamada, On the Lu Qi-keng conjecture, Proc. Amer. Math. Soc. 59 (1976), no. 2, 222–224. MR 425185, DOI 10.1090/S0002-9939-1976-0425185-9
Additional Information
- Harold P. Boas
- Affiliation: Department of Mathematics Texas A&M University College Station Texas 77843–3368
- MR Author ID: 38310
- ORCID: 0000-0002-5031-3414
- Email: boas@math.tamu.edu
- Received by editor(s): December 10, 1994
- Additional Notes: This research was partially supported by NSF grant number DMS-9203514.
- Communicated by: Eric Bedford
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2021-2027
- MSC (1991): Primary 32H10
- DOI: https://doi.org/10.1090/S0002-9939-96-03259-5
- MathSciNet review: 1317032