Minimal Surfaces and Self-shrinkers with Codimension Two in R4

In this talk, we discuss a rigidity result for two dimensional graphical self-shrinker in R⁴. That is a graph of $f(x):R^2\rightarrow R^2$ as a self-shrinker. Our idea is inspired from Mutao Wang’s results of graphical mean curvature flows with arbitrary codimension. If the Jacobian of f is always less than 1, then its graph as a self shrinker is a plane through 0. If time permitted, we also report some recent results in this direction.