The mean curvature flow along the K\"ahler-Ricci flow

Let $(M,\bar{g})$ be a K"ahler surface, and $\mathrm{\Sigma }$ an immersed surface in $M$. The K"ahler angle of $\mathrm{\Sigma }$ in $M$ is introduced by Chern-Wolfson \cite{CW}. Let $(M,\bar{g}(t))$ evolve along the K"ahler-Ricci flow, and ${\mathrm{\Sigma }}_t$ in $(M,\bar{g}(t))$ evolve along the mean curvature flow. We show that the K"ahler angle $\alpha (t)$ satisfies the evolution equation: $$(\frac{\mathrm{\partial }}{\mathrm{\partial }t}-\mathrm{\Delta })\cos \alpha =\mathrm{\mid }\bar{\mathrm{\nabla }}{J}_{{\mathrm{\Sigma }}_t}{\mathrm{\mid }}^2\cos \alpha +R{\sin }^2\alpha \cos \alpha ,$$ where $R$ is the scalar curvature of $(M,\bar{g}(t))$. The equation implies that, if the initial surface is symplectic (Lagrangian), then along the flow, ${\mathrm{\Sigma }}_t$ is always symplectic (Lagrangian) at each time $t$, which we call a symplectic (Lagrangian) K"ahler-Ricci mean curvature flow. In this paper, we mainly study the symplectic K"ahler-Ricci mean curvature flow.