Mean curvature flow of Pinched submanifolds to spheres

The evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of Gerhard Huisken’s seminal paper [Hu1]. More recently, mean curvature flow of higher codimension submanifolds has also received attention. In this paper we prove a result analogous to that of [Hu1] for submanifolds of any codimension. Let F0 : Σn → Rn+k be a smooth immersion of a compact manifold Σ. The mean curvature flow with initial condition F0 is a smooth family of immersions F : Σ× [0,T )→Rn+k satisfying

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