Angle theorems for the Lagrangian mean curvature flow

Abstract

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle $\alpha$ for the corresponding Lagrangian submanifold in the cross product space $L\times M$ satisfies $\text{osc}(\alpha)\le \pi$ . If one considers a 4-dimensional Kahler-Einstein manifold $\overline{M}$ of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that $L\subset\overline{M}$ is a compact oriented Lagrangian submanifold w.r.t. J such that the Kahler form $\overline{\kappa}$ w.r.t.K restricted to L is positive and $\text{osc}(\alpha)\le \pi$ , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. $\overline{\kappa}$ .