In this thesis, we generalize Colding and
Minicozzi''s work on the stability of self-shrinkers in the
hypersurface case to higher co-dimensional cases. The 1st and 2nd
variation formulae of the $F$-functional are derived and an equivalent
condition to the stability in general codimension is found. Using
the equivalent condition, we can classify $F$-stable product
self-shrinkers and show that the Lagrangian self-shrinkers given by
Anciaux are $F$- unstable.
