Bernstein Theorems for Calibrated Submanifolds in $\mathbb{R}^7$ and $\mathbb{R}^8$

TL;DR

This work develops Bernstein-type rigidity for calibrated submanifolds, focusing on coassociative graphs in $\mathbb{R}^7$ and Cayley graphs in $\mathbb{R}^8$. By exploiting the $G_2$ and Spin$(7)$ structures, it encodes the graph slopes through singular values $(\lambda_1,\lambda_2,\lambda_3)$ (or $(\lambda_0,\lambda_1,\lambda_2,\lambda_3)$) and their elementary symmetric polynomials, deriving precise relations like $\lambda_1+\lambda_2+\lambda_3=\lambda_1\lambda_2\lambda_3$ and similar Cayley constraints. The core method reduces the Bernstein problem to the positivity of quadratic forms in the second fundamental form, which is analyzed via block matrices L, $\mathbf{L}_0$, and $\mathbf{M}$, and certified using Bernstein polynomials to obtain explicit region bounds. The main results state that, under quantified conditions on the pairwise products of slopes (or their Cayley analogue) and within components of the singular-value loci, complete graphical coassociative or Cayley submanifolds must be affine, thereby extending rigidity phenomena beyond hypersurfaces to calibrated submanifolds in $\mathbb{R}^7$ and $\mathbb{R}^8$.

Abstract

This paper explores the Bernstein problem of smooth maps $f:\mathbb{R}^4 \to \mathbb{R}^3$ whose graphs form coassociative submanifolds in $\mathbb{R}^7$. We establish a condition, expressed in terms of the second elementary symmetric polynomial of the map's slope, that ensures $f$ is affine. A corresponding result is also established for Cayley submanifolds in $\mathbb{R}^8$.

Figures (3)

• Figure 1: The conditions
• Figure 2: On $\mathcal{G}_0$
• Figure 3: On $\mathcal{G}_+$

Theorems & Definitions (35)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof