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In this paper, we prove convergence of the high codimension mean curvatureflow in the sphere to either a round point or a totally geodesic sphereassuming a pinching condition between the norm squared of the secondfundamental form and the norm squared of the mean curvature and the backgroundcurvature 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 $n=2$ andcodimension $2$, we consider an alternative pinching condition which includesthe normal curvature of the normal bundle. Finally, we sharpen the Chern-doCarmo-Kobayashi curvature condition for surfaces in the four sphere - thiscurvature condition is sharp for minimal surfaces and we conjecture it to besharp for curvature flows in the sphere.

Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow