Reductions of minimal Lagrangian submanifolds with symmetries

Abstract

Let M be a Fano manifold equipped with a Kahler 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 with respect to a globally conformal Kahler metric is equivalent to the minimality of the reduced Lagrangian submanifold $$L_0=L/K$$ in a Kahler quotient $$M_0$$ with respect to the Hsiang–Lawson metric. Furthermore, we give some examples of Kahler reductions by using a circle action obtained from a cohomogenenity one action on a Kahler–Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.