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

TL;DR

This paper studies how immersed surfaces evolve under mean curvature flow within a K"ahler surface that itself evolves via the K"ahler-Ricci flow, focusing on preserving symplectic and Lagrangian properties.

Contribution

It derives the evolution equation for the K"ahler angle during the combined flow and shows the preservation of symplectic and Lagrangian conditions over time.

Findings

The K"ahler angle satisfies a specific evolution equation involving scalar curvature.

Symplectic and Lagrangian properties are preserved along the flow.

Main focus on the behavior of symplectic K"ahler-Ricci mean curvature flow.

Abstract

Let $(M,\overline{g})$ be a K\"ahler surface, and $\Sigma$ an immersed surface in $M$. The K\"ahler angle of $\Sigma$ in $M$ is introduced by Chern-Wolfson \cite{CW}. Let $(M,\overline{g}(t))$ evolve along the K\"ahler-Ricci flow, and $\Sigma_t$ in $(M,\overline{g}(t))$ evolve along the mean curvature flow. We show that the K\"ahler 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) K\"ahler-Ricci mean curvature flow. In this paper, we mainly study the symplectic K\"ahler-Ricci mean curvature flow.