Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 4, pp 915–926 | Cite as

The Schwarz Lemma at the Boundary of the Non-Convex Complex Ellipsoids

  • Le He (何乐)
  • Zhenhan Tu (涂振汊)Email author
Article
  • 32 Downloads

Abstract

Let B2,p:= {z ∈ ℂ2: ∣z12+ ∣z2p < 1} (0 < p< 1). Then, B2,p(0 < p < 1) is a non-convex complex ellipsoid in ℂ2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ∂B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2, p, where z0 is any smooth boundary point of B2,p.

Key words

Boundary Schwarz lemma Holomorphic mappings Kobayashi metric non-convex complex ellipsoids 

2010 MR Subject Classification

32F45 32H02 30C80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Burns D M, Krantz S G. Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J Amer Math Soc, 1994, 7(3): 661–667MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chelst D. A generalized Schwarz lemma at the boundary. Proc Amer Math Soc, 2001, 123(11): 3275–3278MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Franzoni T, Vesentini E. Holomorphic Maps and Invariant Distances. Amsterdan: North-Holland, 1980zbMATHGoogle Scholar
  4. [4]
    Garnett J B. Bounded Analytic Functions. New York: Academic press, 1981zbMATHGoogle Scholar
  5. [5]
    Huang X J. A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains. Can J Math, 1995, 47: 405–420MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Krantz S G. The Schwarz lemma at the boundary. Complex Var Elliptic Equ, 2011, 56(5): 455–468MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Liu T S, Tang X M. Schwarz lemma at the boundary of the Egg Domain B p 1,p2 in ℂn. Canad Math Bull, 2015, 58(2): 381–392MathSciNetCrossRefGoogle Scholar
  8. [8]
    Liu T S, Tang X M. Schwarz lemma at the boundary of strongly pseudoconvex domain in ℂn. Math Ann, 2016, 366: 655–666MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Osserman R. A sharp Schwarz inequality on the boundary. Proc Amer Math Soc, 2000, 128(12): 3513–3517MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Pflug P, Zwonek W. The kobayashi metric for non-convex complex ellipsoids. Complex Var Theory Appl, 1996, 29(1): 59–71MathSciNetzbMATHGoogle Scholar
  11. [11]
    Wang X P, Ren G B. Boundary Schwarz Lemma for Holomorphic Self-mappings of Strongly Pseudoconvex Domains. Complex Anal Oper Theory, 2017, 11: 345–358MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Wu H. Normal families of holomorphic mappings. Acta Math, 1967, 119: 193–233MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

Personalised recommendations