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

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 ${\mathbb{R}}^8$.