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A Hopf Lemma for Holomorphic Maps into Hyperquadrics
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.
Related
- Boundary regularity and Hopf lemma for nondegenerate stable operators
- Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
- The index of conformal Hopf maps
- Holomorphic expanding maps
- Componentwise rigidity of holomorphic isometric maps between bounded symmetric domains with negative conformal factors
BibTeX
@misc{huang2026hopf,
title = {{A Hopf Lemma for Holomorphic Maps into Hyperquadrics}},
author = {Xiaojun Huang and Yuan Zhang and Weixia Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03316},
eprint = {2609.03316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03316},
}
RIS
TY - UNPB
TI - A Hopf Lemma for Holomorphic Maps into Hyperquadrics
AU - Xiaojun Huang
AU - Yuan Zhang
AU - Weixia Zhu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03316
UR - https://arxiv.org/abs/2609.03316
AB - 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.
ER -