The stability of self-shrinkers of mean curvature flow in higher co-dimension

We generalize Colding and Minicozzi's work (2012) on the stability of hypersurface self-shrinkers to higher co-dimension. The first and second variation formulae of the F-functional are derived and an equivalent condition to the stability in general co-dimension is found. We also prove that R-n is the only stable product self-shrinker and show that the closed embedded Lagrangian self-shrinkers constructed by Anciaux are unstable.