Schwarz Lemmas via the Pluricomplex Green’s Function

Abstract

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green’s function. We also prove several other Schwarz lemmas using this function.

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Acknowledgements

I would like to thank Dr. Pranav Haridas for many useful discussions. Prof. Aprameyan Parthasarathy and Dr. Vikramjeet Chandel read parts of this manuscript and made useful suggestions that have greatly improved the manuscript. Prof. G.P. Balakumar patiently answered many questions, both trivial and hard, that were useful in the proofs. I would also like to thank Prof. Autumn Kent for bringing Stanton’s paper [22] to my attention. I am grateful to the anonymous referee for her/his careful reading of our paper and the many useful suggestions that have improved our exposition.

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Correspondence to Jaikrishnan Janardhanan.

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Jaikrishnan Janardhanan is supported by a DST-INSPIRE fellowship from the Department of Science and Technology, India (Grant Number IFA-14/MA-47).

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Janardhanan, J. Schwarz Lemmas via the Pluricomplex Green’s Function. J Geom Anal 30, 4110–4125 (2020). https://doi.org/10.1007/s12220-019-00232-0

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Keywords

  • Schwarz lemma
  • Symmetric product
  • Pluricomplex Green’s function

Mathematics Subject Classification

  • Primary: 32H15
  • 32H20
  • 32U35