  In this paper, we prove convergence of the high codimension mean curvature
flow in the sphere to either a round point or a totally geodesic sphere
assuming a pinching condition between the norm squared of the second
fundamental form and the norm squared of the mean curvature and the background
curvature of the sphere. We show that this pinching is sharp for dimension
$n\geq 4$ but is not sharp for dimension $n=2,3$. For dimension $n=2$ and
codimension $2$, we consider an alternative pinching condition which includes
the normal curvature of the normal bundle. Finally, we sharpen the Chern-do
Carmo-Kobayashi curvature condition for surfaces in the four sphere - this
curvature condition is sharp for minimal surfaces and we conjecture it to be
sharp for curvature flows in the sphere.
