Parallel calibrations and minimal submanifolds

Abstract

Given a parallel calibration $\varphi\in\Omega^{p}(M)$ on a Riemannian manifold $M$, I prove that the $\varphi$-critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $\varphi$-critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^{\infty}(M)$-linear subspace $\mathscr{P}\subset\Omega^{p}(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi–Yau manifolds, (co)associative calibrations on $G_{2}$-manifolds, and the Cayley calibration on $\operatorname{Spin}(7)$-manifolds.)