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

Chun-Kai Lien; Chung-Jun Tsai

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

Differential Geometry 2025-03-24 v1

Authors: Chun-Kai Lien , Chung-Jun Tsai

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 $\mathbb{R}^8$ .

Keywords

Bernstein theorem calibrated submanifolds coassociative Cayley symmetric polynomial

Cite

@article{arxiv.2503.16898,
  title  = {Bernstein Theorems for Calibrated Submanifolds in $\mathbb{R}^7$ and $\mathbb{R}^8$},
  author = {Chun-Kai Lien and Chung-Jun Tsai},
  journal= {arXiv preprint arXiv:2503.16898},
  year   = {2025}
}

Comments

27 pages, 3 figures

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