Singularities of symplectic and Lagrangian mean curvature flows

Abstract

In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Kähler-Einstein surface. We prove that the blow-up flow ${\Sigma }_s^{\infty }$ at a singular point $(X_0,T_0)$ of a symplectic mean curvature flow ${\Sigma }_t$ or of a Lagrangian mean curvature flow ${\Sigma }_t$ is a non trivial minimal surface in ${\mathbf{R}}^4$, if ${\Sigma }_{-\infty }^{\infty }$ is connected.