Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane

TL;DR

This paper studies how Lagrangian spherical surfaces in complex Euclidean space evolve under mean curvature flow, identifying conditions leading to finite-time extinction and convergence to the Clifford torus.

Contribution

It provides a condition on initial Lagrangian tori in c2^2 that guarantees finite-time extinction and convergence to the Clifford torus under mean curvature flow.

Findings

Identifies a specific condition for initial tori leading to finite-time extinction.

Shows convergence of the flow to the Clifford torus after rescaling.

Answers a previously posed question by Neves about Lagrangian torus evolution.

Abstract

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.