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The Exact Comass Criterion for the Tsai--Wang Form
Abstract
For a map F:Ω⊂ℝn→ℝm, let Θ(F) be the n-form on Ω×ℝm introduced by Tsai and Wang [arXiv:2604.04336]. If λ1,…,λr are the nonzero singular values of $d F$ at any fixed point, we prove that Θ(F) has comass one if and only if 𝒮(λ):=∑i=1rλi2/(1+λi2)≤1. Note that if rank d F≤1, 𝒮 is always less than $1$. We also prove a dichotomy that if 𝒮≤ 1, then either 𝒮 < 1 or 𝒮≡1. When 𝒮<1, the graph tangent plane is the only calibrated plane, generalizing the corresponding result for hypersurfaces. When 𝒮≡1, we prove that F is affine or of rank $2$.
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BibTeX
@misc{chen2026exact,
title = {{The Exact Comass Criterion for the Tsai--Wang Form}},
author = {Wei-Chun Chen and Chun-Kai Lien},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39159},
eprint = {2609.39159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39159},
}
RIS
TY - UNPB
TI - The Exact Comass Criterion for the Tsai--Wang Form
AU - Wei-Chun Chen
AU - Chun-Kai Lien
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39159
UR - https://arxiv.org/abs/2609.39159
AB - For a map F:Ω⊂ℝ^n→ℝ^m, let Θ(F) be the n-form on Ω×ℝ^m introduced by Tsai and Wang [arXiv:2604.04336]. If λ_1,…,λ_r are the nonzero singular values of $d F$ at any fixed point, we prove that Θ(F) has comass one if and only if 𝒮(λ):=∑_i=1^rλ_i^2/(1+λ_i^2)≤1. Note that if rank d F≤1, 𝒮 is always less than $1$. We also prove a dichotomy that if 𝒮≤ 1, then either 𝒮 < 1 or 𝒮≡1. When 𝒮<1, the graph tangent plane is the only calibrated plane, generalizing the corresponding result for hypersurfaces. When 𝒮≡1, we prove that F is affine or of rank $2$.
ER -