A note on Schwarz’s lemma

Abstract

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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References

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Acknowledgements

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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Keywords

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

Mathematics Subject Classification

  • 30C80