  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.
