Some remarks on singular capillary cones with free boundary

Abstract

We study singular minimizing capillary cones with free boundary. We provide a stability criterion $\grave{a}$ Jerison-Savin and use it to prove that a minimizing capillary cone is flat when its free boundary mean curvature is non-negative and $n\le 4$, or non-positive and $n\le 6$. We also show that minimizing cones with axially symmetric free boundary are flat in dimensions up to $6$, and derive improved regularity results for graphical minimizing capillary hypersurfaces. The proofs rely on Simons-type inequalities for convex, homogeneous, symmetric functions of the principal curvatures, coupled with a boundary identity intrinsic to the capillary setting.