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$K$aconnectedcompactLiegroupactingon$M$ 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 $L_0=L\mathrm{/}K$ 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.