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 ${\mathcal{C}}^{\infty }(M)$--linear subspace $\sP \subset Ω^p(M)$. 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 $\tSpin(7)$--manifolds.)

v2: substantial revision including new result (Theorem 1.2), 10 pages

References in corpus (1)

• Plurisubharmonicity in a General Geometric Context

Cited by in corpus (2)

• Geometric structures associated with a simple Cartan 3-form
• An Introduction to Potential Theory in Calibrated Geometry