A High-Rank Gap Theorem for Type-I Bounded Symmetric Domains
Abstract
Let Ωr,s and Ωr',s' be Type-I bounded symmetric domains with s>r and s'>r'. Let F:U→ Mr',s'(ℂ) be holomorphic, where Ω̅r,s⊂ U, and suppose that F(Sr,s)⊂ Sr',s'. If k(s−r)≤ s'−r'<(k+1)(s−r) and r'>kr, then, up to target coordinates, the map contains a fixed identity block of size r'−kr. The proof first turns the boundary equation into a matrix identity for the first derivative. A direct dimension count gives g(s−r)≤ s'−r', where g is the rank of a positive semidefinite matrix arising from this identity. The extremal property of the Shilov boundary and the scalar Hopf boundary lemma then show that certain rows of the map are constant. This gives the fixed identity block. We also give examples showing that its size is optimal and explain why the proof needs F on a neighborhood of Ω̅r,s.
Related
- Componentwise rigidity of holomorphic isometric maps between bounded symmetric domains with negative conformal factors
- Rigidity for local holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains
- Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
- Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains
- Quantum harmonic analysis and the Differential operators on weighted Bergman spaces of bounded symmetric domains
BibTeX
@misc{gao2026highrank,
title = {{A High-Rank Gap Theorem for Type-I Bounded Symmetric Domains}},
author = {Yun Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06348},
eprint = {2609.06348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06348},
note = {CC0},
}
RIS
TY - UNPB
TI - A High-Rank Gap Theorem for Type-I Bounded Symmetric Domains
AU - Yun Gao
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06348
UR - https://arxiv.org/abs/2609.06348
AB - Let Ω_r,s and Ω_r',s' be Type-I bounded symmetric domains with s>r and s'>r'. Let F:U→ M_r',s'(ℂ) be holomorphic, where Ω̅_r,s⊂ U, and suppose that F(S_r,s)⊂ S_r',s'. If k(s−r)≤ s'−r'<(k+1)(s−r) and r'>kr, then, up to target coordinates, the map contains a fixed identity block of size r'−kr. The proof first turns the boundary equation into a matrix identity for the first derivative. A direct dimension count gives g(s−r)≤ s'−r', where g is the rank of a positive semidefinite matrix arising from this identity. The extremal property of the Shilov boundary and the scalar Hopf boundary lemma then show that certain rows of the map are constant. This gives the fixed identity block. We also give examples showing that its size is optimal and explain why the proof needs F on a neighborhood of Ω̅_r,s.
ER -