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# Cambridge Journal of Mathematics

## Volume 3 (2015)

### Number 1–2

### Sharp lower bounds on density for area-minimizing cones

Pages: 1 – 18

DOI: http://dx.doi.org/10.4310/CJM.2015.v3.n1.a1

#### Authors

#### Abstract

We prove that the density of a topologically nontrivial, area-minimizing hypercone with an isolated singularity must be greater than $\sqrt{2}$. The Simons’ cones show that $\sqrt{2}$ is the best possible constant. If one of the components of the complement of the cone has nontrivial $k^\textrm{th}$ homotopy group, we prove a better bound in terms of $k$; that bound is also best possible. The proofs use mean curvature flow.

#### Keywords

area-minimizing cone, density, mean curvature flow

#### 2010 Mathematics Subject Classification

Primary 53A10. Secondary 49Q05, 53C44.