Reductions of minimal Lagrangian submanifolds with symmetries

Abstract

Abstract. Let M be a Fano manifold equipped with a Kähler form ω ∈ 2πc 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 Kähler metric is equivalent to the minimality of the reduced Lagrangian submanifold L 0 = L/K in a Kähler quotient M 0 with respect to the Hsiang-Lawson metric. Furthermore, we give some examples of Kähler reductions by using a circle action obtained from a cohomogenenity one action on a Kähler-Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.

On Hamiltonian stable Lagrangian tori in complex hyperbolic spaces

“…Here, it turns out that (π α ) * H ′ α coincides with the mean curvature vector H c of the reduced Lagrangian immersion L/C(K) → CP n−1 (4/ sinh 2 r). This can be shown by using Lemma 3 in [11] and the fact that, in our setting, |ṽ z | gα is constant on L for each α. This proves the second equation of (16).…”