A Construction of New Families of Minimal Lagrangian Submanifolds via Torus Actions

Abstract

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 L_k in M such that all L_k are invariant under Tⁿ⁻¹ and L_k locally converge to L (in particular the supremum of the sectional curvatures of L_k and the distance between L and L_k go to 0 as k ↦ ∞). This result is new even for M = ℂPⁿ for n ≥ 3.