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arXiv · 2609.03316

A Hopf Lemma for Holomorphic Maps into Hyperquadrics

Xiaojun Huang·Yuan Zhang·Weixia Zhu
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Abstract

More than twenty years ago, Baouendi and the first author proved that a holomorphic map between hyperquadrics of the same signature is either totally degenerate or has a nonvanishing normal derivative for its normal component. This established a CR analogue of the classical Hopf lemma in arbitrary codimension, in the absence of pseudoconvexity. They further conjectured that the same Hopf-type property holds for holomorphic maps between Levi-nondegenerate hypersurfaces of the same signature. In this paper, we provide a counterexample to this conjecture in full generality. We also prove the conjecture when the target hypersurface is a hyperquadric of any codimension, arguably the most important case for applications.

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Xiaojun Huang, Yuan Zhang, Weixia Zhu. 2026-09-03. A Hopf Lemma for Holomorphic Maps into Hyperquadrics. https://arxiv.org/abs/2609.03316

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