Abstract
In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain \(D\subset {\mathbb {C}}^m\) into \({\mathbb {P}}^n\) to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto’s work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of \({\mathbb {P}}^n\)-valued holomorphic mappings.
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References
Aladro, G., Krantz, S.G.: A criterion for normality in \({\mathbb{C}}^n\). J. Math. Anal. Appl. 161, 1–8 (1991)
Dethloff, G., Thai, D.D., Trang, P.N.T.: Normal families of meromorphic mappings of several complex variables for moving hypersurfaces in a complex projective space. Nagoya Math. J. 217, 23–59 (2015)
Eremenko, A.: A Picard type theorem for holomorphic curves. Period. Math. Hungar. 38, 39–42 (1999)
Fujimoto, H.: On families of meromorphic maps into the complex projective space. Nagoya Math. J. 54, 21–51 (1974)
Quang, S.D., Tan, T.V.: Normal families of meromorphic mappings of several complex variables into \({\mathbb{C}}P^n\) for moving hypersurfaces. Ann. Polon. Math. 94, 97–110 (2008)
Rutishauser, H.: Über Folgen und Scharen von analytischen und meromorphen Funktionen mehrerer Variabeln, sowie von analytischen Abbildungen. Acta Math. 83, 249–325 (1950)
Stoll, W.: Normal families of non-negative divisors. Math. Z. 84, 154–218 (1964)
Thai, D.D., Trang, P.N.T., Huong, P.D.: Families of normal maps in several complex variables and hyperbolicity of complex spaces. Complex Var. Theory Appl. 48, 469–482 (2003)
Yang, L., Fang, C., Pang, X.: Normal families of holomorphic mappings into complex projective space concerning shared hyperplanes. Pac. J. Math. 272, 245–256 (2014)
Yang, L., Liu, X., Pang, X.: On families of meromorphic maps into the complex projective space. Houst. J. Math. 42, 775–789 (2016)
Acknowledgements
I wish to express my gratitude to Gautam Bharali for helpful discussions in the course of this work. This work is supported by a SERB National Postdoctoral Fellowship (N-PDF) (No. PDF/2017/001140) and a UGC CAS-II grant (No. F.510/25/CAS-II/2018(SAP-I)).
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Communicated by Norman Levenberg.
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Datt, G. Meromorphically Normal Families in Several Variables. Comput. Methods Funct. Theory 22, 307–321 (2022). https://doi.org/10.1007/s40315-021-00413-5
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DOI: https://doi.org/10.1007/s40315-021-00413-5