Recall that the mean curvature flow for an embedded surface \(F_{0 }:\Sigma \rightarrow M\) is described by a one-parameter family of smooth maps \(F_{t }\) that satisfies \({d \over dt} F(x,t)=H(x,t), \;F(x,0)=F_{0}(x),\) where \(H(x,t)\) is the mean curvature vector of \(\Sigma_{t} = F_{t} (\Sigma)\) at \(F(x,t)\) in \(M\). A solution has a Type I singularity at a time \(T>0\) if \(\max_{\Sigma_{t}}|A|^{2}\) becomes unbounded, and \[ \limsup_{t\rightarrow T}(T-t) \max_{\Sigma_{t}}|A|^{2} \leq C, \] where \(A\) is the second fundamental form. The Kähler angle of the initial surface \(\Sigma_{0}\) is determined by the contraction of the Kähler form of \(M\) with the area element of \(\Sigma_{0}\).
The authors study the evolution under mean curvature flow of compact real 2-dimensional symplectic surfaces embedded in Kähler-Einstein surfaces. They prove the following:
Theorem: Consider a compact real 2-dimensional symplectic surface, which is embedded in a Kähler-Einstein surface with nonnegative scalar curvature. If the cosine of the Kähler angle of the initial surface is greater than 0, then the solution of the mean curvature flow does not exhibit a Type I singularity at any time \(T>0\).
If there exists a global solution, then the surface converges to holomorphic curves.
The authors include the evolution equations of the second fundamental form and mean curvature vector for mean curvature flow in arbitrary codimension. The theorem is then proved by contradiction, using a dilation about singularities argument, and a monotonicity formula for \[ \int_{\Sigma_{t}}{1 \over cos(\alpha)}\rho(F,t)\varphi d\mu_{t }, \] where \(\alpha\) is the Kähler angle, \(\rho\) is the backward heat kernel, and \(\varphi\) is a cutoff function.