## A Landau’s theorem in several complex variables.(English)Zbl 1366.32008

One can show, by example, that the Landau theorem and the related result of Brody-Zalcman is not true in several variables.
In this paper, the author introduces a class of holomorphic maps for which one can get a Landau theorem and a Brody-Zalcman theorem in several complex variables.
Denote by $$B^k=\big\{z\in \mathbb C^k : \sum_{j=1}^k|z_j|^2<1\big\}$$ the unit ball in $$\mathbb C^k$$, $$k>1$$. Let $$\Phi: B^k \rightarrow \mathbb C^k$$ be a holomorphic map. Let $$\Phi^\prime(a)$$ denote the Jacobian matrix of $$\Phi$$ computed in the point $$a\in B^k.$$
The author proves the following theorems.
Theorem 2.1. (Brody-Zalcman-type theorem) Let $$C$$ be a positive constant. Consider a family of holomorphic maps $$\Phi$$ satisfying $||\Phi^\prime(a)|| \cdot ||\Phi^\prime(a)^{-1}|| \leq C,\quad \forall a \in B^k.$ If the family is not normal, then, after reparametrization, we can extract a subsequence converging to a non-degenerate holomorphic map $$\Psi : \mathbb C^k \rightarrow \mathbb C^k$$.
Assume that $$\Phi: B^k \rightarrow\mathbb C^k$$, $$\Phi(0) = 0$$, and $$\Phi^\prime(0) = \mathrm{Id}$$.
Theorem 3.1. (Landau-type theorem) Let $$C$$ be a positive constant such that $||\Phi^\prime(z)|| \cdot ||\Phi^\prime(z)^{-1}|| \leq C,\quad \forall z \in B^k.$ Then there exists $$\rho>0$$, depending only on $$C$$, such that $$\Phi(B^k)$$ contains a ball of radius $$\rho>0$$. There exists also a domain $$U$$ such that $$\Phi(U) = B(a,\rho)$$.

### MSC:

 32H99 Holomorphic mappings and correspondences 32A18 Bloch functions, normal functions of several complex variables
Full Text:

### References:

  Bisi, C; Gentili, G, Möbius transformations and the Poincaré distance in the quaternionic setting, Indiana Univ. Math. J., 58, 2729-2764, (2009) · Zbl 1193.30067  Bisi, C., Gentili, G.: On the geometry of the quaternionic unit disc. In: Sabadini, I., Sommen, F. (eds.) Hypercomplex analysis and applications. Trends in Mathematics, pp. 1-11. Springer, Basel (2011) · Zbl 1220.30067  Bisi, C; Stoppato, C, The Schwarz-Pick lemma for slice regular functions, Indiana Univ. Math. J, 61, 297-317, (2012) · Zbl 1271.30024  Bisi, C., Stoppato, C.: Regular vs. classical Möbius transformations of the quaternionic unit ball. In: Gentili, G., Sabadini, I., Shapiro, M., Sommen, F., Struppa, D.C. (eds.) Advances in hypercomplex analysis. Springer INdAM Series, vol. 1, pp. 1-13. Springer, Milan (2013) · Zbl 1270.30018  Bisi, C., Stoppato, C.: Landau’s theorem for slice regular functions on the quaternionic unit ball, Preprint (2016) · Zbl 1368.30023  Brody, R, Compact manifolds and hyperbolicity, Trans. Am. Math. Soc., 235, 213-219, (1978) · Zbl 0416.32013  Chen, H; Gauthier, PM, Bloch constants in several variables, Trans. Am. Math. Soc, 353, 1371-1386, (2001) · Zbl 0966.32002  Conway, J.B.: Functions of One Complex Variable. Graduate Texts in Mathematics, vol. 11. Springer, New York (1973) · Zbl 0277.30001  Duren, P; Rudin, W, Distortion in several variables, Complex Var. Theory Appl., 5, 323-326, (1986) · Zbl 0559.32004  FitzGerald, CH; Gong, S, The Bloch theorem in several complex variables, J. Geom. Anal., 4, 35-58, (1994) · Zbl 0794.32006  Fornaess, JE; Stout, EL, Regular holomorphic images of balls, Ann. Inst. Fourier, 32, 23-36, (1982) · Zbl 0452.32008  Graham, I., Kohr, G.: Geometric function Theory in One and Higher Dimensions. Pure and Applied Mathematics, vol. 255. Dekker, New York (2003) · Zbl 1042.30001  Graham, I; Varolin, D, Bloch constants in one and several variables, Pac. J. Math., 174, 347-357, (1996) · Zbl 0885.32004  Hahn, KT, Higher dimensional generalizations of the Bloch constant and their lower bounds, Trans. Am. Math. Soc., 179, 263-274, (1973) · Zbl 0267.32012  Harris, LA, On the size of balls covered by analytic transformations, Mon. Math., 83, 9-23, (1977) · Zbl 0356.46046  Heins M.: Selected topics in the classical theory of functions of a complex variable. Athena Series: Selected Topics in Mathematics Holt. Rinehart and Winston, New York, (1962) · Zbl 1226.30001  Landau, E, Über die blochsche konstante und zwei verwandte weltkonstanten, Math. Z, 30, 608-634, (1929) · JFM 55.0770.03  Landau, E.: Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. Springer, Berlin (1929) · JFM 55.0171.03  Liu, XY, Bloch functions of several complex variables, Pac. J. Math., 152, 347-363, (1992) · Zbl 0770.32003  Miniowitz, R, Normal families of quasimeromorphic mappings, Proc. Am. Math. Soc., 84, 35-43, (1982) · Zbl 0478.30024  Schiff, J.L.: Normal families. Universitext. Springer, New York (1993) · Zbl 0770.30002  Sibony, N.: Dynamique des applications rationnelles de $$\mathbb{P}^k$$. (French) [Dynamics of rational maps of $$\mathbb{P}^k$$] Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthéses, vol. 8, Soc. Math. France, Paris (1999), pp. 97-185 (1997) · Zbl 1020.37026  Takahashi, S, Univalent mappings in several complex variables, Ann. Math., 53, 464-471, (1951) · Zbl 0044.30804  Wu, H, Normal families of holomorphic mappings, Acta Math., 119, 193-233, (1967) · Zbl 0158.33301  Zalcman, L, A heuristic principle in complex function, Am. Math. Mon., 82, 813-818, (1975) · Zbl 0315.30036  Zalcman, L, Normal families: new perspectives, Bull. (New Series) Am. Math. Soc., 35, 215-230, (1998) · Zbl 1037.30021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.