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 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 \(\overline{\kappa}\) 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. \(\overline{\kappa}\).
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Received: 11 April 2001 / Published online: 29 April 2002
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Smoczyk, K. Angle theorems for the Lagrangian mean curvature flow. Math Z 240, 849–883 (2002). https://doi.org/10.1007/s002090100402
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DOI: https://doi.org/10.1007/s002090100402