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Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
Abstract
We prove a rigidity theorem: a globally symmetric complex Finsler space $(M, J, F)$ is necessarily a Kähler-Berwald space, namely F must be a Kähler-Berwald metric. We also obtain a Schwarz lemma for holomorphic mappings f from an arbitrary bounded symmetric domain D into itself whenever D is endowed with an Aut(D)-invariant Kähler-Berwald metric F such that its holomorphic sectional curvature is bounded below and above by negative constants −K1<0 and −K2<0, respectively. The novelty of this Schwarz lemma is that the Lu constant of (D,F) is optimal for each given F whenever rank(D)≥ 2.
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BibTeX
@misc{he2026schwarz,
title = {{Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics}},
author = {Yong He and Chunping Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34435},
eprint = {2609.34435},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34435},
}
RIS
TY - UNPB
TI - Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
AU - Yong He
AU - Chunping Zhong
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34435
UR - https://arxiv.org/abs/2609.34435
AB - We prove a rigidity theorem: a globally symmetric complex Finsler space $(M, J, F)$ is necessarily a Kähler-Berwald space, namely F must be a Kähler-Berwald metric. We also obtain a Schwarz lemma for holomorphic mappings f from an arbitrary bounded symmetric domain D into itself whenever D is endowed with an Aut(D)-invariant Kähler-Berwald metric F such that its holomorphic sectional curvature is bounded below and above by negative constants −K_1<0 and −K_2<0, respectively. The novelty of this Schwarz lemma is that the Lu constant of (D,F) is optimal for each given F whenever rank(D)≥ 2.
ER -