Schwarz Lemma at the Boundary on the Classical Domain of Type $$\mathcal{III}$$ I I
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Schwarz Lemma at the Boundary on the Classical Domain of Type III ∗ Taishun LIU1
Xiaomin TANG2
Wenjun ZHANG3
Abstract Let RIII (n) be the classical domain of type III with n ≥ 2. This article is devoted to a deep study of the Schwarz lemma on RIII (n) via not only exploring the smooth boundary points of RIII (n) but also proving the Schwarz lemma at the smooth boundary point for holomorphic self-mappings of RIII (n). Keywords Holomorphic mapping, Schwarz lemma at the boundary, The classical domain of type III 2000 MR Subject Classification 32H02, 32H99, 30C80
1 Introduction Schwarz lemma is one of the most important results in the classical complex analysis. A great deal of work has been devoted to generalizations of Schwarz lemma to more general settings. We refer to [1–8] for a more complete insight on the Schwarz lemma. In the case of several complex variables, the Schwarz lemma originated from the work of Cartan. In [9], Cartan obtained the following rigidity theorem for holomorphic mappings. Theorem 1.1 (cf. [9]) Let Ω be a bounded domain in Cn . If f : Ω → Ω is a holomorphic mapping such that f (z) = z + o(kz − z0 k) as z → z0 for some z0 ∈ Ω, then f (z) ≡ z. On the other hand, Look first considered the properties of the Jacobian matrix of holomorphic mapping in [10]. Theorem 1.2 (cf. [10]) Let Ω be a bounded domain in Cn , and let f be a holomorphic self-mapping of Ω which fixes a point p ∈ Ω. Then the eigenvalues of Jf (p) all have modulus not exceeding 1 and | det Jf (p)| ≤ 1. Moreover, if | det Jf (p)| = 1, then f is a biholomorphism of Ω. It is natural to explore the high-dimensional versions of the Schwarz lemma at the boundary. Motivated by Theorem 1.1, Burns and Krantz first studied the boundary Schwarz lemma and Manuscript received May 29, 2018. of Mathematics, Huzhou University, Huzhou 313000, Zhejiang, China. E-mail: [email protected] 2 Corresponding author. Department of Mathematics, Huzhou University, Huzhou 313000, Zhejiang, China. E-mail: [email protected] 3 Department of Mathematics, Shenzhen University, Shenzhen 518060, Guangdong, China. E-mail: [email protected] ∗ This work was supported by the National Natural Science Foundation of China (Nos. 11571105, 11771139). 1 Department
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T. S. Liu, X. M. Tang and W. J. Zhang
the rigidity problem for holomorphic mappings in [11]. See [12–16] for more on these matters. Motivated by Theorem 1.2, we focused on the characterizations of the Jacobian matrix of holomorphic mapping at the boundary point of some domains in Cn , and established the boundary Schwarz lemmas (see [17–18]). These results are widely applied in many fields. By the classical Schwarz lemma at the boundary, Bonk improved the Bloch constant in [19], and Liu, Ren, Gong and Zhang obtained the growth, covering and distortion theorems for biholomorphic convex mappings or quasi-convex mappings on some domains in [20–22]. Recently, using the Schwarz lemma at the boundary of the u
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