The mean curvature flow along the Kähler-Ricci flow

Abstract

Let $(M,\overline{g})$ be a Kähler surface, and $\Sigma$ an immersed surface in $M$. The Kähler angle of $\Sigma$ in $M$ is introduced by Chern-Wolfson \cite{CW}. Let $(M,\overline{g}(t))$ evolve along the Kähler-Ricci flow, and ${\Sigma }_t$ in $(M,\overline{g}(t))$ evolve along the mean curvature flow. We show that the Kähler angle $\alpha (t)$ satisfies the evolution equation: $$(\frac{\partial }{\partial t}-\Delta )\cos \alpha =|\overline{\nabla }{J}_{{\Sigma }_t}{|}^2\cos \alpha +R{\sin }^2\alpha \cos \alpha ,$$ 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, ${\Sigma }_t$ 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.

23 pages

References in corpus (1)

• Scalar Curvature Behavior for Finite Time Singularity of Kähler-Ricci Flow