The Mean Curvature of First-Order Submanifolds in Exceptional Geometries with Torsion

TL;DR

This paper derives formulas for the mean curvature of special submanifolds in G2 and Spin(7) geometries with torsion, providing criteria for minimality and new obstructions to their existence.

Contribution

It introduces formulas for mean curvature in torsionful exceptional geometries and characterizes structures where these submanifolds are minimal.

Findings

Formulas for mean curvature of associative, coassociative, and Cayley submanifolds with torsion

Characterization of G2 and Spin(7) structures with minimal submanifolds

New obstructions to local existence of coassociative 4-folds in torsionful G2-structures

Abstract

We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative 4-fold, Cayley 4-fold) is a minimal submanifold. In the process, we obtain new obstructions to the local existence of coassociative 4-folds in G2-structures with torsion.