Compactness, Finiteness Properties of Lagrangian Self-shrinkers in R^4 and Piecewise Mean Curvature Flow

In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R⁴. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R⁴, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in R³ by Colding and Minicozzi.