Cambridge Journal of Mathematics

Volume 3 (2015)

Number 1–2

Sharp lower bounds on density for area-minimizing cones

Pages: 1 – 18



Tom Ilmanen (Departement Mathematik, ETH Zentrum, Zürich, Switzerland)

Brian White (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


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.


area-minimizing cone, density, mean curvature flow

2010 Mathematics Subject Classification

Primary 53A10. Secondary 49Q05, 53C44.

Published 5 June 2015