Flow of Symplectic Surfaces

Geometers have been interested in constructing minimal surfaces for long time. One possible way to produce such surfaces is to deform a given surface by its mean curvature vector. More precisely, the surface evolves in the gradient flow of the area functional, and such a flow is the so-called mean curvature flow. A mean curvature flow, however, develops singularity after finite time in general [H2]. It is therefore desirable to understand behavior of the flow near the singular points (cf. [B], [CL], [E1-E2], [H1-H3], [HS1-HS2], [I1-I2], [Wa1], [Wh1-Wh2] and so on). We consider this problem, in this paper, for compact symplectic surfaces moving by mean curvature flow in a Kähler-Einstein surface. It was observed in [CT2] that symplectic surface remains symplectic along the flow (also see [CL], [Wa1]), and the flow has long time existence in the graphic case as discussed in [CLT] and [Wa2]. One of our motivations of considering symplectic surfaces is inspired by the symplectic isotopic problem for symplectic surfaces in Del Pezzo surfaces, i.e., those complex surfaces with positive first Chern class. It was conjectured in [T] that every embedded orientable closed symplectic surface in a compact Kähler-Einstein surface is isotopic to a symplectic minimal surface in a suitable sense. When Kähler-Einstein surfaces are of positive scalar curvature, this was proved for lower degrees by using pseudo-holomorphic curves (cf. [ST], [Sh]). It would be interesting to have a proof of this result, even for lower degrees, by using the mean curvature flow. In the negative