Abstract
The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forelli’s theorem on the complex analyticity of the functions that are: (i) \({\mathcal {C}}^\infty \) smooth at a point, and (ii) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol’d, V.I.: Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, Berlin (1983)
Chirka, E.M.: Variations of Hartogs’ theorem. Proc. Steklov Inst. Math. 253, 212–224 (2006)
Forelli, F.: Pluriharmonicity in terms of harmonic slices. Math. Scand. 41(2), 358–364 (1977). (MR0477146 (57 #16689))
Greene, R.E., Kim, K.T., Krantz, S.G.: The Geometry of Complex Domains. Birkhäuser-Verlag, Boston (2011)
Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations. Graduate Studies in Mathematics, vol. 86. American Mathematical Society (2007)
Joo, J.-C., Kim, K.-T., Schmalz, G.: A generalization of Forelli’s theorem. Math. Ann. 355(3), 1171–1176 (2013)
Kim, K.-T., Poletsky, E.A., Schmalz, G.: Functions holomorphic along holomorphic vector fields. J. Geom. Anal. 19(3), 655–666 (2009)
Rao, M., Stetkær, H.: Complex analysis. An invitation. A Concise Introduction to Complex Function Theory. World Scientific Publishing Co., Inc, Teaneck (1991)
Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824 (1957)
Stoll, W.: The characterization of the strictly parabolic manifolds. Ann. Scuola Norm. Sup. Pisa 7, 87–154 (1980)
Ueda, T.: Normal forms of attracting holomorphic maps. Math. J. Toyama Univ. 22, 25–34 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of the J.-C. Joo and K.-T. Kim are supported in part by Grant 2011-007831 of the National Research Foundation of Korea. K.-T. Kim is also supported in part by the Grant 2011-0030044 (The SRC-GAIA) of the NRF of Korea. G. Schmalz is supported by ARC Discovery Grant DP130103485.
Rights and permissions
About this article
Cite this article
Joo, JC., Kim, KT. & Schmalz, G. On the generalization of Forelli’s theorem. Math. Ann. 365, 1187–1200 (2016). https://doi.org/10.1007/s00208-015-1277-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1277-x