Abstract

One of the broad aims of differential geometry is to classify manifolds with a curvature condition, a famous example is Hamilton’s classification of three manifolds with positive Ricci curvature, the seminal result which led to the Ricci flow resolution of Thurston’s geometrization conjecture. Other results in this vein are the classification of four manifolds with positive isotropic curvature in dimension four by Hamilton and two -convex hypersurfaces of Euclidean space of dimension greater than four but Huisken-Sinestrari.

In this talk, we will consider the mean curvature flow in the sphere with a quadratic curvature condition that generalizes the two-convexity condition introduced by Huisken-Sinestrari. We classify type I solutions and show that the class of such submanifolds is closed under connected sum. Finally, we classify type II singularities using convexity type estimates for mean curvature flow in the sphere.