Abstract: We consider curvature flows of hypersurfaces in a Riemannian manifold. An important example is the flow by mean curvature, but for many geometric problems flows by other speeds have proven to be a more effective tool. The key step for most applications is to understand the structure of singularities. We will discuss a new a priori 'convexity' estimate, which implies that singularities are positively curved, even if the initial datum is highly non-convex. As a consequence, we obtain a complete classification of Type I singularities under very general conditions, despite having no analogue of Huisken's monotonicity formula for mean curvature flow.

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