The space of compact self-shrinking solutions to the Lagrangian Mean Curvature Flow in $\mathbb C^2$

Let $F_n :(\Sigma, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$. We show that a subsequence of $\{F_n\}$ converges smoothly to a branched conformally immersed Lagrangian self-shrinker $F_\infty : (\Sigma, h)\to \mathbb C^2$. When the area bound is less than $16\pi$, the limit $F_\infty$ is an embedded torus. When the genus of $\Sigma$ is one, we can drop the assumption on convergence $h_n\to h$. When the genus of $\Sigma$ is zero, we show that there is no branched immersion of $\Sigma$ as a Lagrangian shrinker, generalizing the rigidity result of Smoczyk in dimension two by allowing branch points.