Sharp Lower Bounds on Density of Area-Minimizing Cones

We prove that the density of a topologically nontrivial, area-minimizing hypercone with an isolated singularity must be greater than the square root of 2. The Simons' cones show that this is the best possible constant. If one of the components of the complement of the cone has nontrivial kth homotopy group, we prove a better bound in terms of k; that bound is also best possible. The proofs use mean curvature flow.