In this paper we investigate connections between minimal Lagrangian submanifolds and holomorphic vector fields in Kähler manifolds. Our main result is: Let M2n (n ≥ 2) be a Kähler-Einstein manifold with positive scalar curvature with an effective, structure-preserving action by an n-torus Tn. Then precisely one regular orbit L of the Tn-action is a minimal Lagrangian submanifold of M. Moreover there is an (n − 1)-torus Tn−1 ⊂ Tn and a sequence of non-flat immersed minimal Lagrangian tori Lk in M such that all Lk are invariant under Tn−1 and Lk locally converge to L (in particular the supremum of the sectional curvatures of Lk and the distance between L and Lk go to 0 as k ↦ ∞). This result is new even for M = ℂPn for n ≥ 3.
"A Construction of New Families of Minimal Lagrangian Submanifolds via Torus Actions." J. Differential Geom. 58 (2) 233 - 261, June, 2001. https://doi.org/10.4310/jdg/1090348326