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MiS Preprint
31/2001
Angle theorems for the Lagrangian mean curvature flow
Knut Smoczyk
Abstract
We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive 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 $osc (\alpha) \leq \pi$. If one considers a 4-dimensional Kähler-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 Kähler form $\overline{K}$ w.r.t. K restricted to L is positive, then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. $\overline{K}$.
