PreprintarXiv licencearXiv
Schwarz lemmas and rigidity of ∂̅b-harmonic maps
Abstract
We prove Schwarz lemmas for ∂̅b-harmonic maps from complete pseudo-Hermitian manifolds into K"ahler manifolds of negative curvature. For maps of bounded generalized dilatation we bound the full and the horizontal differential. For CR maps a coupled Bochner identity yields rank-dependent horizontal estimates and a bound for the Reeb differential. On Sasakian sources and complex hyperbolic targets we classify the maps attaining the horizontal bound, bound the deviation from equality in integral form by the deficit at one point, and prove compactness of almost extremal sequences. As an application, a Siu-type divergence identity yields foliation, pluriharmonicity and rank rigidity under volume growth conditions.
Related
- Target Geometry in Prescribed-Value Schwarz Lemmas for Harmonic Maps
- A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry
- Spectral and convex geometry of α-harmonic maps and sharp Morse index rigidity
- Spectral Schwarz Lemmas and Parallel Obstructions to Metric Comparison
- Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
BibTeX
@misc{huang2026schwarz,
title = {{Schwarz lemmas and rigidity of \(\bar\partial_b\)-harmonic maps}},
author = {Xin Huang and Hui Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05181},
eprint = {2610.05181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05181},
}
RIS
TY - UNPB
TI - Schwarz lemmas and rigidity of ∂̅_b-harmonic maps
AU - Xin Huang
AU - Hui Liu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05181
UR - https://arxiv.org/abs/2610.05181
AB - We prove Schwarz lemmas for ∂̅_b-harmonic maps from complete pseudo-Hermitian manifolds into K"ahler manifolds of negative curvature. For maps of bounded generalized dilatation we bound the full and the horizontal differential. For CR maps a coupled Bochner identity yields rank-dependent horizontal estimates and a bound for the Reeb differential. On Sasakian sources and complex hyperbolic targets we classify the maps attaining the horizontal bound, bound the deviation from equality in integral form by the deficit at one point, and prove compactness of almost extremal sequences. As an application, a Siu-type divergence identity yields foliation, pluriharmonicity and rank rigidity under volume growth conditions.
ER -