The mean curvature flow along the Kähler-Ricci flow

Abstract

Let $(M,\overline{g})$ be a Kähler surface, and $Σ$ an immersed surface in $M$. The Kähler angle of $Σ$ in $M$ is introduced by Chern-Wolfson \cite{CW}. Let $(M,\overline{g}(t))$ evolve along the Kähler-Ricci flow, and $Σ_t$ in $(M,\overline{g}(t))$ evolve along the mean curvature flow. We show that the Kähler angle $α(t)$ satisfies the evolution equation: $$ (\frac{\partial}{\partial t}-Δ)\cosα=|\overline\nabla J_{Σ_t}|^2\cosα+R\sin^2α\cosα, $$ where $R$ is the scalar curvature of $(M, \overline{g}(t))$. The equation implies that, if the initial surface is symplectic (Lagrangian), then along the flow, $Σ_t$ is always symplectic (Lagrangian) at each time $t$, which we call a symplectic (Lagrangian) Kähler-Ricci mean curvature flow. In this paper, we mainly study the symplectic Kähler-Ricci mean curvature flow.