  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.
