Special Lagrangian Submanifolds (Slag) are volume-minimizing submanifolds in Calabi--Yau manifolds, and they are important objects in Mirror Symmetry. However, the general existence theorem for Slags is still unknown. One potential scheme for finding Slags is the Lagrangian Mean Curvature Flow (LMCF), which is the negative gradient flow of the volume functional.

In this talk, I will first introduce the basic notions of Slags and LMCF and then describe some recent results.