Regularity of minimal surfaces near quadratic cones


Hardt-Simon proved that every area-minimizing hypercone $\mathbf {C}$ having only an isolated singularity fits into a foliation of $\mathbb R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf {C}$. In this paper we prove that if a stationary integral $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb R^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons’ cone $\mathbf {C}^{3,3}$), then $\mathrm {spt}\, M \cap B_{1/2}$ is a $C^{1,\alpha }$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.


Nick Edelen

University of Notre Dame, Notre Dame, IN

Luca Spolaor

University of California, San Diego, La Jolla, CA