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De Rham Decomposition Theorem for Strongly Convex Kähler–Berwald Manifolds

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Abstract

In this paper, we prove that the unit ball \(B_n\subset \mathbb {C}^n\) admits no \(\text{ Aut }(B_n)\)-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the canonical Poincaré–Bergman metric, while the unit polydisk \(P_n\subset \mathbb {C}^n(n\ge 2)\) admits infinite many \(\text{ Aut }(P_n)\)-invariant strongly convex complex Finsler metrics other than the Bergman metric. The \(\text{ Aut }(P_n)\)-invariant strongly pseudoconvex complex Finsler metrics (which are not necessary Hermitian quadratic) are explicitly constructed and are proved to be strongly convex Kähler–Berwald metrics. We also investigate the existence of holomorphic invariant strongly pseudoconvex non-Hermitian quadratic complex Finsler metrics on reducible complex manifolds, and give a de Rham decomposition theorem for strongly convex Kähler–Berwald manifolds.

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Acknowledgements

The author would like to express his sincere thanks to the referees for their valuable suggestions and comments.

Funding

The author is supported by the National Natural Science Foundation of China (Grant Number 12071386 and 11671330).

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Chunping Zhong

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Zhong, C. De Rham Decomposition Theorem for Strongly Convex Kähler–Berwald Manifolds. Results Math 78, 25 (2023). https://doi.org/10.1007/s00025-022-01797-z

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