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Burns-Krantz rigidity in non-smooth domains

Part of: Holomorphic functions of several complex variables

Published online by Cambridge University Press:  18 March 2025

Włodzimierz Zwonek Open the ORCID record for Włodzimierz Zwonek [Opens in a new window]
Włodzimierz Zwonek*
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, Kraków 30–348, Poland
*
e-mail: [email protected]
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Abstract

Motivated by recent papers [11] and [19] we prove a boundary Schwarz lemma (Burns-Krantz rigidity type theorem) for non-smooth boundary points of the polydisc and symmetrized bidisc. Basic tool in the proofs is the phenomenon of invariance of complex geodesics (and their left inverses) being somehow regular at the boundary point under the mapping satisfying the property as in the Burns-Krantz rigidity theorem that lets the problem reduce to one dimensional problem. Additionally, we make a discussion on bounded symmetric domains and suggest a way to prove the Burns-Krantz rigidity type theorem in these domains that however cannot be applied for all bounded symmetric domains.

Keywords

Burns-Krantz rigidity propertypolydiscsymmetrized bidiscLempert theorem

MSC classification

Primary: 32A40: Boundary behavior of holomorphic functions
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 10
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Research was partially supported by the Sheng grant no. 2023/48/Q/ST1/00048 of the National Science Center, Poland.

References

Agler, J., Lykova, Z., and Young, N. J., Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc . Mem. Amer. Math. Soc. 258(2019), 108 pp.Google Scholar
Agler, J., Lykova, Z., and Young, N. J., Characterizations of some domains via carathéodory extremals . J. Geom. Anal. 29(2019), 3039–3054.CrossRefGoogle Scholar
Agler, J. and Young, N. J., A Schwarz lemma for the symmetrized bidisc . Bull. London Math. Soc. 33(2001), 175–186.CrossRefGoogle Scholar
Agler, J. and Young, N. J., The hyperbolic geometry of the symmetrized bidisc . J. Geom. Anal. 14(2004), 375–403.CrossRefGoogle Scholar
Agler, J. and Young, N. J., The complex geodesics of the symmetrized bidisc . Int. J. Math. 17(2006), 375–391.CrossRefGoogle Scholar
Bracci, F., Kosiński, Ł., and Zwonek, W., Slice rigidity property of holomorphic maps Kobayashi isometrically preserving complex geodesics . J. Geom. Anal. 31(2021), no. 2011, 11292–11311.CrossRefGoogle Scholar
Burns, D. and Krantz, S. G., Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary . J. Amer. Math. Soc. 7(1994), no. 3, 661–676.CrossRefGoogle Scholar
Chang, C.-H., Hu, M. C., and Lee, H.-P., Extremal analytic discs with prescribed boundary data . Trans. Amer. Math. Soc. 310(1988), 355–369.CrossRefGoogle Scholar
Costara, C., The symmetrized bidisc and Lempert’s theorem . Bull. London Math. Soc. 36(2004), 656–662.CrossRefGoogle Scholar
Edigarian, A., Kosiński, Ł., and Zwonek, W., The Lempert theorem and the tetrablock . J. Geom. Anal. 23(2013), 1818–1831.CrossRefGoogle Scholar
Fornaess, J. E. and Rong, F., The boundary rigidity for holomorphic self-maps of some fibered domains . Math. Res. Lett. 28(2021), 697–706.CrossRefGoogle Scholar
Ghosh, G. and Zwonek, W., $2$ -proper holomorphic images of classical Cartan domains . Indiana Univ. Math. J. (2023), to appear.Google Scholar
Huang, X., A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains . Can. J. Math. 47(1995), 405–420.CrossRefGoogle Scholar
Huang, X., A preservation principle of extremal mappings near a strongly pseudoconvex point and its applications . Illinois J. Math. 38(1994), 283–302.CrossRefGoogle Scholar
Jarnicki, M. and Pflug, P., Invariant distances and metrics in complex analysis. 2nd extended ed., de Gruyter Expositions in Mathematics, 9, Walter de Gruyter, Berlin, Boston, 2013, xvii+861 pp.CrossRefGoogle Scholar
Kosiński, Ł. and Zwonek, W., Nevanlinna–Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock . J. Geom. Anal. 26(2016), 1863–1890.CrossRefGoogle Scholar
Lempert, L., La métrique de Kobayashi et la représentation des domaines sur la boule . Bull. Soc. Math. Fr. 109(1981), 427–474.CrossRefGoogle Scholar
Lempert, L., Intrinsic distances and holomorphic retracts . In: L. Iliev and V. Andreev (eds.), Complex analysis and applications ’81 (Varna, 1981), Publishing House of Bulgarian Academy of Sciences, Sofia, 1984, pp. 341–364.Google Scholar
Ng, S.-C. and Rong, F., The Burns-Krantz rigidity on bounded symmetric domains . Bull. London Math. Soc. 57(2025), 115–119.CrossRefGoogle Scholar
Pflug, P. and Zwonek, W., Description of all complex geodesics in the symmetrized bidisc . Bull. London Math. Soc. 37(2005), 575–584.CrossRefGoogle Scholar
Zimmer, A., Two boundary rigidity results for holomorphic maps . Amer. J. Math. 144(2022), 119–168.CrossRefGoogle Scholar