Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in $mathbb{C}^n$

摘要

We give new examples of self-shrinking and self-expanding Lagrangian solutions to the Mean Curvature Flow (MCF). These are Lagrangian submanifolds in $mathbb{C}^n$ , which are foliated by (n − 1)-spheres (or more generally by minimal (n − 1)-Legendrian submanifolds of $mathbb{S}^{2n-1}$ ), and for which the study of the self-similar equation reduces to solving a non-linear Ordinary Differential Equation (ODE). In the self-shrinking case, we get a family of submanifolds generalising in some sense the self-shrinking curves found by Abresch and Langer.