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

TL;DR

This paper studies the limits of sequences of conformally immersed Lagrangian self-shrinkers in ^2, showing convergence to branched self-shrinkers and classifying limits under area and genus constraints.

Contribution

It proves convergence of sequences of Lagrangian self-shrinkers with bounded area to branched limits and classifies the possible limits based on area and genus, extending previous rigidity results.

Findings

Sequences with area less than 16\u03a0 converge to embedded tori.

No branched Lagrangian shrinkers exist for genus zero surfaces.

Convergence results hold without metric assumptions for genus one surfaces.

Abstract

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.