Abstract
We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.
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Acknowledgements
We are very grateful to Professor Jiaping Wang for helpful suggestions and his support. We want to express our gratitude to Professors Kefeng Liu, Lei Ni and Fangyang Zheng for useful communications and discussions. We also thank Professor Jianfei Wang for helpful communications and the warm hospitality when we visited Huaqiao University in July 2021 and thank Professor Yibin Ren for some nice discussions.
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Dedicated to Professor Peter Li on the occasion of his 70th birthday.
This research is partially supported by NSFC Grants Nos. 11901530, 11801516, 12071161 and Zhejiang Provincial NSF Grant No. LY19A010017.
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Chen, H., Nie, X. Schwarz Lemma: The Case of Equality and an Extension. J Geom Anal 32, 92 (2022). https://doi.org/10.1007/s12220-021-00771-5
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DOI: https://doi.org/10.1007/s12220-021-00771-5