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 K\"ahler-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 ${\bf R}^4$, if $\Sigma_{-\infty}^\infty$ is connected.