Abstract
Domains that are increasing union of balls (up to biholomorphism) and on which the Kobayashi metric vanishes identically arise inexorably in complex analysis. In this article, we show that in higher dimensions these domains have infinite volume and the Bergman spaces of these domains are trivial. As a consequence they fail to be strictly pseudo-convex at each of their boundary points although these domains are pseudo-convex by definition. These domains can be of different types and one of them is Short \(\mathbb {C}^k\)’s. In pursuit of identifying the Runge Short \(\mathbb {C}^k\)’s (up to biholomorphism), we introduce a special class of Short \(\mathbb {C}^k\)’s, called Loewner Short \(\mathbb {C}^k\)’s. These are those Short \(\mathbb {C}^k\)’s which can be exhausted in a continuous manner by a strictly increasing parametrized family of open sets, each of which is biholomorphically equivalent to the unit ball and therefore, they are Runge up to biholomorphism. Although, the question of whether all Short \(\mathbb {C}^k\)’s are Runge (up to biholomorphism), or whether all Short \(\mathbb {C}^k\)’s are Loewner remains unsettled, we show that the typical Short \(\mathbb {C}^k\)’s are Loewner. In the final section, we construct a bunch of non-autonomous basins of attraction, which serve as interesting examples of Short \(\mathbb {C}^2\)’s.
Similar content being viewed by others
References
Abbondandolo, A., et al.: A survey on non-autonomous basins in several complex variables. arxiv:1311.3835
Arosio, L., Bracci, F., Wold, E.F.: Solving the Loewner PDE in complete hyperbolic starlike domains of \(\mathbb{C}^n\). Adv. Math. 242, 209–216 (2013)
Arosio, L., Boc-Thaler, L., Peters, H.: A transcendental Hénon map with an oscillating wandering Short \(\mathbb{C}^2\). Math. Z. (2021). https://doi.org/10.1007/s00209-020-02677-4
Behnke, H., Stein, K.: Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität. Math. Ann. 116, 204–216 (1939)
Bera, S.: Examples of non-autonomous basins of attraction-II. J. Ramanujan Math. Soc. 34(3), 343–363 (2019)
Bera, S., Pal, R., Verma, K.: Examples of non-autonomous basins of attraction. Illinois J. Math. 61, 531–567 (2017)
Fornæss, J.E.: An increasing sequence of Stein manifolds whose limit is not Stein. Math. Ann. 223, 275–277 (1976)
Fornæss, J.E.: A counterexample for the Levi problem for branched Riemann domains over \(\mathbb{C}^n\). Math. Ann. 234(3), 275–277 (1978)
Fornæss, J.E.: Short \(\mathbb{C}^k\), Complex analysis in several variables-Memorial Conference of Kiyoshi Oka’s Centennial Birthday, 95–108, Adv. Stud. Pure Math., 42. Math. Soc. Japan, Tokyo (2004)
Fornæss, J.E., Sibony, N.: Increasing sequences of complex manifolds. Math. Ann. 255, 351–360 (1981)
Fornaess, J.E., Stout, E.L.: Polydiscs in complex manifolds. Math. Ann. 227, 145–153 (1977)
Forstneric, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of \(\mathbb{C}^n\). Invent. Math. 112, 323–349 (1993)
Gallagher, A.-K., Harz, T., Herbort, G.: On the dimension of the Bergman space for some unbounded domains. J. Geom. Anal. 27, 1435–1444 (2017)
Jucha, P.: A remark on the dimension of the Bergman space of some Hartogs domains. J. Geom. Anal. 22, 23–37 (2009)
Pal, R., Verma, K.: Ergodic properties of families of Hénon maps. Ann. Polon. Math. 121(1), 45–71 (2018)
Peter, P., Włodzimierz, Z.: \(L_h^2\)-Functions in unbounded balanced domains. J. Geom. Anal. 27, 2118–2130 (2017)
Thaler, L.B., Forstneric, F.: A long \(\mathbb{C}^2\) without holomorphic functions. Anal. PDE 9(8), 2031–2050 (2016)
Wiegerinck, J.J.O.O.: Domains with finite dimensional Bergman space. Math. Z. 187, 559–562 (1984)
Wold, E.F.: A Fatou-Bieberbach domain in \(\mathbb{C}^2\) which is not Runge. Math. Ann. 340(4), 775–780 (2008)
Wold, E.F.: A long \(\mathbb{C}^2\) which is not Stein. Ark. Mat. 48(1), 207–210 (2010)
Acknowledgements
The present article was initiated during the 2020 Complex Dynamics conference at the CIRM-Luminy, France. Both authors are grateful to CIRM for providing local hospitality during the conference. The second author would like to thank Koushik Ramachandran and Sivaguru Ravisankar for partially supporting her travel to CIRM. The second author was supported by National Board of Higher Mathematics postdoctoral fellowship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fornæss, J.E., Pal, R. Notes on the Short \({\mathbb {C}}^k\)’s. J Geom Anal 32, 133 (2022). https://doi.org/10.1007/s12220-022-00869-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-022-00869-4