Some remarks on singular capillary cones with free boundary

TL;DR

This work advances the understanding of minimizing capillary cones with free boundary by establishing a Jerison–Savin–style stability framework tailored to capillarity. It leverages a generalized Simons inequality for convex, symmetric, homogeneous functions of the principal curvatures and a boundary inequality to derive rigidity results, notably showing that in dimension $n=4$ a minimizing capillary cone with $H_{∂M}$ of constant sign is flat, and providing an alternative proof of nonexistence of singular cones in $n=3$. The authors also demonstrate that axially symmetric free boundaries are flat for $n<7$ and obtain $n^*_G(θ) ≥5$ for graphical cones, yielding improved singular set bounds for capillary drops via Federer dimension reduction. The results illuminate a deep link between capillarity problems and the one-phase Bernoulli problem, highlighting how curvature-based stability controls translate into global rigidity in low dimensions, with potential impact on regularity theory for free boundary problems in capillarity contexts.

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.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.3: The Caffarelli-Jerison-Kenig stability inequality
• Proposition 2.4
• proof