Minimal Lagrangian tori in Kahler Einstein manifolds

TL;DR

This paper constructs minimal Lagrangian submanifolds in Kahler-Einstein manifolds with positive scalar curvature using torus actions, revealing a unique minimal orbit and a sequence of approximating tori.

Contribution

It demonstrates the existence and approximation of minimal Lagrangian tori in Kahler-Einstein manifolds with torus symmetries, highlighting a new geometric construction.

Findings

Existence of a unique minimal Lagrangian orbit under T^n-action.

Construction of a sequence of non-flat minimal Lagrangian tori converging to the orbit.

Approximation of minimal Lagrangian submanifolds via invariant tori.

Abstract

In this paper we use structure preserving torus actions on Kahler-Einstein manifolds to construct minimal Lagrangian submanifolds. Our main result is: Let N^2n be a Kahler-Einstein manifold with positive scalar curvature with an effective T^n-action. Then precisely one regular orbit L of the T-action is a minimal Lagrangian submanifold of N. Moreover there is an (n-1)-torus T^n-1 in T^n and a sequence of non-flat immersed minimal Lagrangian tori L_k in N, invariant under T^n-1 s.t. L_k locally converge to L (in particular the supremum of the sectional curvatures of L_k and the distance between L_k and L go to 0 as k goes to infinity.