  We study geometric properties of the Lagrangian self-shrinking tori in
$\mathbb R^4$. 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 {\L}ojasiewicz-Simon type gradient
inequality for the branched conformal self-shrinking tori and then combining
with the compactness theorem in \cite{CMa}. When the area bound is small, we
show that any Lagrangian self-shrinking torus in $\mathbb R^4$ with small area
is embedded with uniform curvature estimates, and the space of such tori is
compact.
  Using the finiteness of entropy values, we construct a piecewise Lagrangian
mean curvature flow for Lagrangian immersed tori in $\mathbb R^4$, 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 $\mathbb R^3$ in \cite{CM}.
