The Exact Comass Criterion for the Tsai--Wang Form

Abstract

For a map $F:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^m$, let $\Theta (F)$ be the $n$-form on $\Omega \times {\mathbb{R}}^m$ introduced by Tsai and Wang [arXiv:2604.04336]. If ${\lambda }_1,\dots ,{\lambda }_r$ are the nonzero singular values of $dF$ at any fixed point, we prove that $$\Theta (F)\text{ has comass one if and only if }\mathcal{S}(\lambda ):=\sum _{i=1}^r\frac{{\lambda }_i^2}{1+{\lambda }_i^2}\le 1.$$ Note that if $\text{rank}dF\le 1$, $\mathcal{S}$ is always less than $1$. We also prove a dichotomy that if $\mathcal{S}\le 1$, then either $\mathcal{S}<1$ or $\mathcal{S}\equiv 1$. When $\mathcal{S}<1$, the graph tangent plane is the only calibrated plane, generalizing the corresponding result for hypersurfaces. When $\mathcal{S}\equiv 1$, we prove that $F$ is affine or of rank $2$.

arXiv comment13 pages