Reductions of minimal Lagrangian submanifolds with symmetries

Let $M$ be a Fano manifold equipped with a K\"ahler form $\omega\in 2\pi c_1(M)$ and $K$ a connected compact Lie group acting on $M$ as holomorphic isometries. In this paper, we show the minimality of a $K$-invariant Lagrangian submanifold $L$ in $M$ w.r.t. a globally conformal K\"ahler metric is equivalent to the minimality of the reduced Lagrangian submanifold $L_0=L/K$ in a K\"ahler quotient $M_0$ w.r.t. the Hsiang-Lawson metric. Furthermore, we give some examples of K\"ahler reductions by using a circle action obtained from a cohomogenenity one action on a K\"ahler-Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.