Calibrated fibrations on noncompact manifolds via group actions

Abstract

In this paper we use Lie group actions on noncompact Riemannian manifolds with calibrations to construct calibrated submanifolds. In particular, if we have an $(n-1)$-torus acting on a noncompact Calabi-Yau $n$-fold with a trivial first cohomology, then we have a special Lagrangian fibration on that $n$-fold. We produce several families of examples for this construction and give some applications to special Lagrangian geometry on compact almost Calabi-Yau manifolds. We also use group actions on noncompact $G_2$-manifolds to construct coassociative submanifolds, and we exhibit some new examples of coassociative submanifolds via this setup.