"Introduction to Lagrangian Mean Curvature Flow"

Abstract: In the last few years, the mean curvature flow in higher codimension has attracted special attention.
Together with the graphs and simplectic submanifolds, Lagrangian submanifolds can be said to be one of the best known classes regarding mean curvature flow. A reason for this increasing interest is that the Lagrangian condition is preserved by mean curvature flow. We are interested in the Euclidean ambient space, where either the flow has an eternal solution or it fails to exist after a finite time, giving rise to a singularity. A natural question is to understand the geometric and analytic nature of these singularities.
In order to understand Type I singularities, we will deal with the study of those Lagrangian surfaces in complex Euclidean plane C^2 which evolve by homotheties of the ambient space: the so-called self-similar solutions. In this context, we will discuss the classification of the Hamiltonian stationary self-similar solutions to the Lagrangian mean curvature flow in C^2, as well as some rigidity results for the Clifford torus in the class of compact self-shrinkers for the Lagrangian mean curvature flow.