preprint

Inserted: 12 feb 2025
Last Updated: 12 feb 2025

Year: 2025

Abstract:

We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion à 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.