Parallel calibrations and minimal submanifolds

Details Parallel calibrations and minimal submanifolds Parallel calibrations and minimal submanifolds

2008-08-15

arxiv.org/abs/0808.2158v4

Mathematics

Differential Geometry

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

Scientific paper

Given a parallel calibration $\phi \in \Omega^p(M)$ on a Riemannian manifold $M$, I prove that the $\phi$--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $\phi$--critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^\infty(M)$--linear subspace $\sP \subset \Omega^p(M)$. 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 $\tSpin(7)$--manifolds.)

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