Calibrated fibrations on noncompact manifolds via group actions (Q1847877)

Let \(M\) be a Riemannian manifold and \(\xi\) a closed \(k\)-form on \(M\). \(\xi\) is said to be a calibration on \(M\) if for any oriented \(k\)-plane \(\kappa\) in the tangent bundle of \(M\), \(\xi|_\kappa\leq \text{vol}(\kappa)\). An oriented \(k\)-dimensional submanifold \(L\) of \(M\) may be said to be calibrated by \(\xi\) if \(\xi\) restricts to the volume form on \(L\). It is known that such a manifold is homologically volume minimizing. In this paper the author applies Lie group actions on noncompact manifolds with calibrations to construct these submanifolds. In the particular case where an \((n-1)\)-torus acts on a noncompact Calabi-Yau \(n\)-fold with trivial first cohomology, he obtains a special Lagrangian fibration and exhibits several examples. He also uses group actions on noncompact \(G_2\)-manifolds for constructing coassociative submanifolds and numerous examples are presented.