Sharp lower bounds on density for area-minimizing cones (Q496563)

In this paper, the authors consider the question for the infimum for density among all area-minimizing hypercones \(C \subset \mathbb{R}^{n}\) with an isolated singularity at the origin. This question is equivalent to the question of the infimum for density among all pairs \((M,x)\), where \(M\) is an area-minimizing hypersurface in a Riemannian manifold and \(x\) is an interior singular point of \(M\). Specifically, they provide a sharp answer to this question under a topological assumption, which amounts to a positive answer to a conjecture of Bruce Solomon in the case when one considers the (particularly prevalent examples of) hypercones that satisfy the assumption: Theorem. Suppose that \(C \subset \mathbb{R}^{n}\) is an area-minimizing hypercone with an isolated singularity at the origin, and also that at least one of the two components of \(\mathbb{R}^{n} \setminus C\) is non-contractible. Then the density of \(C\) at the origin is greater than \(\sqrt{2}\). An equivalent formulation for minimal submanifolds of spheres is given. The proof relies on several facts about mean curvature flow.