Sharp Lower Bounds on Density of Area-Minimizing Cones

TL;DR

This paper establishes sharp lower bounds on the density of area-minimizing cones with isolated singularities, using mean curvature flow, and demonstrates the optimality of these bounds through examples like Simons' cones.

Contribution

It provides the first sharp lower bounds on the density of topologically nontrivial area-minimizing cones, extending to bounds involving homotopy groups.

Findings

Density of nontrivial area-minimizing cones exceeds √2

Bounds are optimal, as shown by Simons' cones

Improved bounds for cones with nontrivial homotopy groups

Abstract

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.