Mean curvature flow and minimal surfaces

In the first part, we show that for an ancient solution of the mean curvature flow, each time slice Mt is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family Mt that evolves under mean curvature flow for all negative time t.; In the second part, we construct a metric with positive scalar curvature on S2xS 1 and a sequence of embedded minimal tori converging to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. This is a joint work with Darren Lee.