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)

 
 

 

Estimates for invariant metrics on $ \mathbb{C}$-convex domains


Authors: Nikolai Nikolov, Peter Pflug and Włodzimierz Zwonek
Journal: Trans. Amer. Math. Soc. 363 (2011), 6245-6256
MSC (2010): Primary 32F45, 32A25
DOI: https://doi.org/10.1090/S0002-9947-2011-05273-6
Published electronically: June 27, 2011
MathSciNet review: 2833552
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Geometric lower and upper estimates are obtained for invariant metrics on $ \mathbb{C}$-convex domains containing no complex lines.


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

  • 1. M. Andersson, M. Passare, R. Sigurdsson, Complex convexity and analytic functionals, Birkhäuser, Basel-Boston-Berlin, 2004. MR 2060426 (2005a:32011)
  • 2. E. Bedford, S. I. Pinchuk, Convex domains with noncompact groups of automorphisms, Sb. Math. 82 (1995), 1-20. MR 1275970 (95e:32037)
  • 3. S. Blumberg, Das Randverhalten der Bergman-Kerns und der Bergman-Metrik auf lineal konvexe Gebieten endlichen Typs, Dissertation, Universität Wuppertal, 2005.
  • 4. H. P. Boas, E. Straube, On equality of line type and variety type of real hypersurfaces in $ \mathbb{C}^n$, J. Geom. Anal. 2 (1992), 95-98. MR 1151753 (93g:32024)
  • 5. H. P. Boas, E. Straube, J. Yu, Boundary limits of the Bergman kernel and metric, Michigan Math. J. 42 (1995), 449-461. MR 1357618 (96j:32029)
  • 6. J.-H. Chen, Estimates of invariant metrics on convex domains, Ph.D. Thesis, Purdue University, 1989.
  • 7. M. Conrad, Nicht isotrope Abschätzungen für lineal konvexe Gebiete endlichen Typs, Dissertation, Universität Wuppertal, 2002.
  • 8. K. Diederich, G. Herbort, Pseudoconvex domains of semiregular type, Contributions to complex analysis and analytic geometry (H. Skoda and J. M. Trepreau, eds.), Aspects of Math. E26, Vieweg, Braunschweig, 1994, pp. 127-162. MR 1319347 (96b:32019)
  • 9. K. Diederich, J. E. Fornaess, Lineally convex domains of finite type: holomorphic support functions, manuscripta math. 112 (2003), 403-431. MR 2064651 (2005g:32011)
  • 10. T. W. Gamelin, Uniform algebras and Jensen measures, Cambridge Univ. Press, Cambridge-New York, 1978. MR 521440 (81a:46058)
  • 11. T. Hefer, Hölder and $ L^p$ estimates for $ \overline{\partial}$ on convex domains of finite type depending on Catlin's multitype, Math, Z. 242 (2002), 367-398. MR 1980628 (2004e:32041)
  • 12. T. Hefer, Extremal bases and Hölder estimates for $ \overline{\partial}$ on convex domains of finite type, Mich. Math. J. 52 (2004), 573-602. MR 2097399 (2005f:32065)
  • 13. G. Herbort, On the Bergman metric near a plurisubharmonic barrier point, Prog. Math. 188 (2000), 123-132. MR 1782663 (2001g:32079)
  • 14. L. Hörmander, Notions of convexity, Birkhäuser, Basel-Boston-Berlin, 1994. MR 1301332 (95k:00002)
  • 15. D. Jacquet, $ \mathbb{C}$-convex domains with $ C^2$ boundary, Complex Variables and Elliptic Equations 51 (2006), 303-312. MR 2218722 (2007j:32005)
  • 16. M. Jarnicki, P. Pflug, Invariant distances and metrics in complex analysis, de Gruyter, Berlin-New York, 1993. MR 1242120 (94k:32039)
  • 17. L. Lee, Asymptotic behavior of the Kobayashi metric on convex domains, Pacific J. Math. 238 (2008), 105-118. MR 2443509 (2009j:32009)
  • 18. L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. MR 660145 (84d:32036)
  • 19. M. Lieder, Das Randverhalten der Kobayashi und Carathéodory-Metrik auf lineal konvexe Gebieten endlichen Typs, Dissertation, Universität Wuppertal, 2005.
  • 20. J. D. McNeal, Convex domains of finite type, J. Funct. Anal. 108 (1992), 361-373. MR 1176680 (93h:32020)
  • 21. J. D. McNeal, Estimates on the Bergman Kernels of Convex domains, Adv. Math. 109 (1994), 108-139. MR 1302759 (95k:32023)
  • 22. J. D. McNeal, Invariant metric estimates for $ \bar\partial$ on some pseudoconvex domains, Ark. Mat. 39 (2001), 121-136. MR 1821085 (2002a:32040)
  • 23. N. Nikolov, Nontangential weighted limit of the infinitesimal Carathéodory metric in an $ h$-extendible boundary point of a smooth bounded pseudoconvex domain in $ \mathbb{C}^n$, Acta Math. Hung. 82 (1999), 311-324. MR 1675619 (2000b:32060)
  • 24. N. Nikolov, Localization of invariant metrics, Arch. Math. 79 (2002), 67-73. MR 1923040 (2003f:32014)
  • 25. N. Nikolov, P. Pflug, Behavior of the Bergman kernel and metric near convex boundary points, Proc. Amer. Math. Soc. 131 (2003), 2097-2102. MR 1963755 (2004a:32007)
  • 26. N. Nikolov, P. Pflug, Estimates for the Bergman kernel and metric of convex domains in $ \mathbb{C}^n$, Ann. Polon. Math. 81 (2003), 73-78. MR 1977762 (2004b:32003)
  • 27. N. Nikolov, P. Pflug, P. J. Thomas, W. Zwonek, On a local characterization of pseudoconvex domains, Indiana Univ. Math. J. 58 (2009), to appear.
  • 28. N. Nikolov, P. Pflug, W. Zwonek, An example of a bounded $ \mathbb{C}$-convex domain which is not biholomorphic to a convex domain, Math. Scand. 102 (2008), 149-155. MR 2420684 (2009b:32014)
  • 29. A. Noell, Peak points in boundaries not of finite type, Pacific J. Math. 123 (1986), 385-390. MR 840849 (87i:32023)
  • 30. N. Sibony, Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987), 299-319. MR 894582 (88g:32036)
  • 31. J. Yu, Multitypes of convex domains, Indiana Univ. Math. J. 41 (1992), 837-849. MR 1189914 (93i:32023)
  • 32. J. Yu, Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J. 43 (1994), 1271-1295. MR 1322619 (96b:32016)
  • 33. J. Yu, Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains, Trans. Amer. Math. Soc. 347 (1995), 587-614. MR 1276938 (95d:32028)
  • 34. J. Yu, Singular Kobayashi metrics and finite type conditions, Proc. Amer. Math. Soc. 123 (1995), 121-130. MR 1231046 (95c:32025)
  • 35. S. V. Znamenskiĭ, L. N. Znamenskaya, Projective convexity in $ \mathbb{C}\mathbb{P}^n$, Siberian Math. J. 38 (1997), 685-698. MR 1474912 (98k:52002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32F45, 32A25

Retrieve articles in all journals with MSC (2010): 32F45, 32A25


Additional Information

Nikolai Nikolov
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Email: nik@math.bas.bg

Peter Pflug
Affiliation: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany
Email: peter.pflug@uni-oldenburg.de

Włodzimierz Zwonek
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Łojasiewicza 6, 30-348 Kraków, Poland
Email: Wlodzimierz.Zwonek@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-2011-05273-6
Keywords: $\mathbb C$-convex domain, Carathéodory, Kobayashi and Bergman metrics, Bergman kernel
Received by editor(s): December 15, 2008
Received by editor(s) in revised form: September 16, 2009
Published electronically: June 27, 2011
Additional Notes: This paper was written during the stay of the first-named author at the Carl von Ossietzky Universität Oldenburg (November-December 2008) supported by the DFG grant 436POL113/103/0-2. The third-named author was supported by the research grant No. N N201 361436 of the Polish Ministry of Science and Higher Education.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society