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arXiv · 2412.02553

RC-positivity, Schwarz's lemma and comparison theorems

Zhiyao Xiong·Xiaokui Yang·Shing-Tung Yau
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Abstract

It is well-known that the classical Schwarz lemma yields an explicit comparison of two Hermitian metrics with uniform constant negative curvature bounds through holomorphic maps between complex manifolds. In this paper, we establish Schwarz lemmas for holomorphic bundle maps between abstract Hermitian holomorphic vector bundles with various positive curvature bounds. As applications, we prove Schwarz lemmas for holomorphic maps between complex manifolds whose curvature tensors are described by the notion ``RC-positivity''. In particular, new diameter and volume comparison theorems are obtained by using Schwarz lemmas.

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Zhiyao Xiong, Xiaokui Yang, Shing-Tung Yau. 2024-12-03. RC-positivity, Schwarz's lemma and comparison theorems. https://arxiv.org/abs/2412.02553

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