Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration

1. Problem Statement and Context

The paper addresses the regularity and minimality properties of free-boundary surfaces (interfaces between a liquid and a container) within nonsmooth domains, specifically circular cones in high dimensions.

• Mathematical Setting: The problem considers a liquid volume $v$ inside a container $\Omega \subset \mathbb{R}^{n+1}$. The configuration minimizes the free-energy functional $P(F; \Omega) + \gamma P(F; \partial\Omega)$, where $P$ denotes the relative perimeter. The paper focuses on the case $\gamma = 0$, which corresponds to minimizing the relative perimeter $P(F; \Omega)$ subject to a volume constraint.
• The Specific Question: In a circular cone $\Omega_\lambda$ with opening angle $\alpha$, is the intersection of the cone with an axial hyperplane (a hyperplane containing the cone's axis) an area-minimizing surface with respect to free-boundary variations?
• The Conflict: Previous work by the author and G.P. Leonardi established a "Vertex-skipping Theorem" for $n=3$ (in $\mathbb{R}^3$), stating that the boundary of an almost minimizer in a convex domain cannot contain isolated singularities (vertices) of the container's boundary. This implies that for $n=3$, axial hyperplanes are unstable in narrow cones. The paper investigates whether this "vertex-skipping" property holds in higher dimensions ($n \geq 4$).

2. Methodology: Free-Boundary Calibration

The core methodology is an adaptation of the calibration method, originally introduced by Bombieri, De Giorgi, and Giusti for minimal surfaces.

• Standard Calibration: A calibration is a closed differential form (or a divergence-free vector field) that bounds the volume form of a surface, proving its minimality.
• Free-Boundary Adaptation: In this context, the calibration vector field $Z$ must satisfy specific boundary conditions:
  1. Tangency: $Z$ must be tangent to the boundary of the cone $\partial \Omega_\lambda$. This ensures that variations of the surface within the cone do not incur energy penalties from the boundary term.
  2. Alignment: On the candidate minimal surface (the axial hyperplane), $Z$ must align perfectly with the surface's normal.
  3. Boundedness: The norm of $Z$ must be $\leq 1$ everywhere.
• Construction Strategy:
  1. Step 1 (Boundary Calibration): Construct a divergence-free vector field $Y$ on the boundary of the cone $S_\lambda = \partial \Omega_\lambda \setminus \{0\}$ that calibrates the intersection of the axial hyperplane with the boundary ($S^0_\lambda$). This reduces to solving a specific differential inequality involving a function $h(u)$ derived from the geometry of the cone.
  2. Step 2 (Extension): Extend $Y$ from the boundary $S_\lambda$ into the interior of the cone $\Omega_\lambda$ via vertical projection to create the global vector field $Z$. The paper proves that due to the specific metric structure of the cone, this extension preserves the divergence-free property ($\text{div } Z = 0$).
• Technical Challenge: The vector field is not defined on the entire closed cone (specifically, it is undefined on a 2-dimensional subset where $x'=0$). The author employs an approximation argument (removing a small neighborhood of the singular set and taking the limit) to apply the Divergence Theorem rigorously.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

For dimensions $n \geq 4$, there exists a threshold parameter $\bar{\lambda}(n)$ defined as:
$$\bar{\lambda}(n) = \frac{1}{2} \frac{n-3}{\sqrt{n-2}}$$
If the cone parameter $\lambda$ (related to the opening angle $\alpha$ by $\lambda = \cot(\alpha/2)$) satisfies $0 < \lambda \leq \bar{\lambda}(n)$ (i.e., the cone is wide enough), then the axial hyperplane $H_\lambda$ is area-minimizing with respect to free-boundary variations inside the cone.

Dimensional Gap and Counterexample

• Counterexample to Vertex-Skipping: The result provides a definitive counterexample to the validity of the Vertex-skipping Theorem in dimensions $n \geq 4$. In these high dimensions, the singularity at the vertex of the cone (the origin) can be part of the singular set of a minimizer if the cone is sufficiently wide.
• The Case $n=3$: The result does not apply to $n=3$. The threshold $\bar{\lambda}(3)$ would be zero (or undefined in the same sense), consistent with previous findings that axial hyperplanes are unstable in 3D cones.
• Dimensional Bound on Singularities: The paper suggests that the bound on the Hausdorff dimension of the singular set of the free-boundary, previously estimated as $\dim_H(\Sigma \cap \partial \Omega) \leq n-3$ in polyhedral settings, cannot be improved beyond $n-4$ in general nonsmooth domains. Specifically, for $n=4$, the origin is a singular point of the minimizer.

Technical Construction

The paper explicitly constructs the calibration vector field $Z$ by:

1. Defining a function $h(u)$ based on the ansatz $h(u) = \cos(\beta(\arctan u)) / \cos(\arctan u)$.
2. Solving a differential inequality for $\beta(\theta)$ to ensure the calibration condition $|\omega_h|_g \leq 1$ holds.
3. Proving that a solution exists if and only if $n \geq 4$ and $\lambda \leq \bar{\lambda}(n)$.

4. Significance

• Resolution of High-Dimensional Regularity: The paper clarifies the behavior of free-boundary minimal surfaces in nonsmooth domains in high dimensions, showing a distinct phase transition compared to the 3D case.
• Limitations of Regularity Theory: It demonstrates that regularity results relying on "vertex-skipping" (where minimizers avoid the singular points of the container) are dimension-dependent and fail for $n \geq 4$ in wide cones.
• Methodological Advancement: It successfully adapts the classical calibration method to a free-boundary setting with a non-smooth boundary, handling the technical difficulties of vector fields defined on subsets of the domain and the singularity at the cone's vertex.
• Implications for Geometric Measure Theory: The findings refine the understanding of the Hausdorff dimension of singular sets in capillarity problems, suggesting that the bound $n-4$ is sharp for certain nonsmooth geometries.

In summary, Vianello proves that in high dimensions ($n \geq 4$), wide circular cones admit axial hyperplanes as free-boundary area-minimizers, thereby disproving the universality of vertex-skipping behavior and establishing a precise dimensional threshold for this phenomenon.