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

Let $F_n:(\mathrm{\Sigma },h_n)\to {\mathbb{C}}^2$ 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 $F_{\mathrm{\infty }}:(\mathrm{\Sigma },h)\to {\mathbb{C}}^2$. When the area bound is less than $16\pi$, the limit $F_{\mathrm{\infty }}$ is an embedded torus. When the genus of $\mathrm{\Sigma }$ is one, we can drop the assumption on convergence $h_n\to h$. When the genus of $\mathrm{\Sigma }$ is zero, we show that there is no branched immersion of $\mathrm{\Sigma }$ as a Lagrangian shrinker, generalizing the rigidity result of Smoczyk in dimension two by allowing branch points.