A note on Schwarz’s lemma


We present a maximum principle for metrics with negative curvature. This principle is essentially a reformulation of the Minda–Schober proof of Ahlfors’s theorem, which by itself is a version of Schwarz’s lemma in differential geometry language. This maximum principle leads to the concept of extremal metrics.

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  1. 1.

    Ahlfors, L.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43(3), 359–364 (1938)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Beardon, A., Minda, D.: The Hyperbolic Metric and Geometric Function Theory. Quasiconformal Mappings and Their Applications, pp. 9–56. Narosa, New Delhi (2007)

    MATH  Google Scholar 

  3. 3.

    Heins, M.: On a class of conformal metrics. Nagoya Math. J. 21, 1–60 (1962)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Minda, D., Schober, G.: Another elementary approach to the theorems of Landau, Montel, Picard and Schottky. Complex Var. theory Appl. 2(2), 157–164 (1983)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Minda, D.: The strong form of Ahlfors’ lemma. Rocky Mt. J. Math. 17(3), 457–461 (1987)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Schwarz, H.: Gesammelte mathematische Abhandlungen. Band I, II. Chelsea, Bronx (1972). Nachdruck in einem Band der Auflage von 1890

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The author gratefully thanks to the referee for the constructive comments and recommendations which helped to improve the quality of the paper.

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Correspondence to Javad Mashreghi.

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This work was partially supported by NSERC Discovery Grant, Canada.

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Mashreghi, J. A note on Schwarz’s lemma. Complex Anal Synerg 7, 8 (2021). https://doi.org/10.1007/s40627-021-00077-w

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  • Schwarz’s lemma
  • SK-metric
  • Laplacian
  • curvature

Mathematics Subject Classification

  • 30C80