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

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.

arXiv comment14 pages

References in corpus (1)

• A canonical way to deform a Lagrangian submanifold