Mean curvature flow of surfaces in a hyperk\"ahler $4$-manifold

Abstract

In this paper, we firstly prove that every hyper-Lagrangian submanifold $L^{2n} (n > 1)$ in a hyperk\"ahler $4n$-manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the complex phase map of self-shrinking surfaces in $\mathbb{R}^4$. Last but not least, by using the previous rigidity result, we show that the mean curvature flow from a closed surface with the image of the complex phase map contained in $\mathbb{S}^2\setminus\overline{\mathbb{S}}^{1}_{+}$ in a hyperk\"ahler $4$-manifold does not develop any Type \Rmn{1} singularity.