Some remarks on singular capillary cones with free boundary

TL;DR

This paper investigates the stability and flatness of singular capillary cones with free boundaries, providing new criteria and results for dimensions up to 6, based on curvature inequalities and boundary conditions.

Contribution

It introduces a Jerison-Savin style stability criterion for capillary hypersurfaces and proves flatness of certain minimizing cones in low dimensions, advancing understanding of capillary cone regularity.

Findings

Minimizing cones with non-sign-changing mean curvature are flat in dimensions up to 4.

Non-trivial axially symmetric cones are unstable in dimensions up to 6.

A Simons-type inequality is established for convex, homogeneous, symmetric functions of principal curvatures.

Abstract

We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $\`a$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing cones with non-sign-changing mean curvature are flat. We apply this criterion to minimizing capillary drops and, additionally, establish the instability of non-trivial axially symmetric cones in dimensions up to $6$. The main results are based on a Simons-type inequality for a class of convex, homogeneous, symmetric functions of the principal curvatures, combined with a boundary condition specific to the capillary setting.