Non-flat Minimal Capillary Cones

• Non-flat minimal capillary cones are singular area-minimizing hypersurfaces that meet boundaries at fixed contact angles, leading to genuine geometric singularities.
• They are rigorously characterized through variational formulations and classification results that establish sharp dimensional thresholds for the onset of non-flat behavior.
• Their stability analysis and explicit construction in high dimensions provide practical insights into boundary regularity theory for both local and nonlocal capillarity functionals.

Non-flat minimal capillary cones are singular area-minimizing hypersurfaces in a domain, invariant under dilations, meeting the boundary (typically a hyperplane or the surface of a solid cone) at a fixed contact angle. These objects are central in the boundary regularity theory for capillary surfaces, free-boundary minimal surfaces, and related variational problems, including local and nonlocal (fractional) capillarity functionals. Non-flatness here indicates that the cone’s cross-section in the unit sphere (its "link") is not a flat equator or planar region, so genuine geometric singularities arise. Their existence, uniqueness, classification, and stability properties delineate sharp threshold dimensions and symmetries for boundary singularities in regularity theory.

1. Definitions and Variational Formulations

A minimal capillary cone is a $1$-homogeneous set $E \subset \mathbb{R}^{n+1}$ with boundary $M = \partial E \cap \{x_{n+1} \geq 0\}$, minimizing a capillarity-modified area functional

$$\mathscr{F}_\theta(E) = \mathcal{H}^n(\partial E \cap \{x_{n+1} > 0\}) - \cos \theta \; \mathcal{H}^n(\partial E \cap \{x_{n+1}=0\})$$

for contact angle $\theta \in (0,\pi)$. The minimization is over compact perturbations preserving the boundary condition that $M$ meets the "container" hyperplane $\{x_{n+1}=0\}$ at the prescribed angle: $$\cos\theta = \langle v, n_{\partial D} \rangle \quad \text{on} \; \Gamma = M \cap \partial D,$$ with $v$ the unit normal to $M$ and $n_{\partial D}$ the inward normal of the container. If $M$ is smooth, this yields the capillarity problem Euler–Lagrange system: $$H_M = 0\quad\text{and}\quad \cos\theta = \langle v, n_{\partial D}\rangle \;\text{along}\; \Gamma.$$ For graphical cones over $\mathbb{R}^n$, the PDE reads $$M(u)=\mathrm{div}(\nabla u/\sqrt{1+|\nabla u|^2})=0$$ in $\{u>0\}$ and $|\nabla u|=\tan\theta$ at the boundary.

In fractional capillarity, the nonlocal capillary energy involves a double integral kernel, with the minimizer’s extension solving a local Dirichlet problem in one higher dimension, and boundary contacts prescribed via adhesion coefficients or fractional Young’s law (Dipierro et al., 2020).

2. Classification in Low and Intermediate Dimensions

Recent work has established a rigidity theory eliminating non-flat minimal capillary cones in low dimensions under natural geometric constraints. Using a Jerison–Savin type stability criterion and a Simons-type inequality adapted to boundary settings, one finds:

• Flatness in $n\leq 4$ for non-sign-changing mean curvature: Any minimizing capillary cone with non-sign-changing $H_\Gamma$ at the boundary is flat (i.e., a half-hyperplane) (Pacati et al., 11 Feb 2025).
• Instability of nontrivial axially symmetric cones for $n\leq 6$: If the cross-section is axially symmetric, flatness persists up to dimension $6$ (Pacati et al., 11 Feb 2025).
• Graphical cones: In $n=4$, all minimizing capillary cones that are graphical over $\{x_{n+1}=0\}$ are flat. The minimal dimension for possible singular cones in capillarity, $n^*(\theta)$, thus satisfies $n^*(\theta)\geq 5$.
• Singular set dimension bound: Free boundary minimizers in $\mathbb{R}^n$ have singular set of Hausdorff dimension at most $n-5$ in the capillary Bernoulli problem, improving the previous $n-7$ bound (Pacati et al., 11 Feb 2025).
• Circular cones: In $\mathbb{R}^3$, all capillary minimal surfaces in a solid circular cone are planar disks; no non-flat solutions exist (López et al., 2014).

These results deeply constrain possible singular behaviors for capillary problems in dimensions $n\leq 6$.

3. Existence and Structure in High Dimensions

Non-flat strictly area-minimizing capillary cones first arise in high dimensions, specifically in $n\geq 7$. Recent constructions in (Firester et al., 26 Jan 2026) produce a complete continuum of non-flat minimizing capillary cones $C_{n,k,\theta}$ invariant under $O(n-k)\times O(k)$ symmetry. The key construction principle is the reduction to a one-dimensional ODE for a profile function $f(t)$ in cylindrical or spherical variables: $$u(x,y) = \rho f(t), \quad t = \frac{|y|}{\sqrt{|x|^2+|y|^2}}$$ with $f$ solving a specific nonlinear second-order equation with capillary boundary data.

The solutions interpolate between singular one-phase cones (as $\theta\to 0$) and halved Lawson cones (as $\theta\to \pi/2$). Monotonicity and uniqueness for the ODE, combined with sub-/super-solution methods, establish strict minimizing properties for $n\geq 7$. These examples show, for the first time, that in $n=7$ there exist non-flat minimizing capillary boundary cones, confirming that boundary singularities in capillarity can occur in codimension $7$ (Firester et al., 26 Jan 2026).

The following table summarizes critical flatness vs. existence thresholds:

Dimension	Graphical / Mean Curvature Condition	Non-flat Cones Possible?
$n\leq 4$	$H_\Gamma$ non-sign-changing	No
$n\leq 6$	Axial symmetry	No
$n\geq 7$	General	Yes

4. Explicit Cones in Sliding and Free-Boundary Plateau Problems

Boundary minimal cones with sliding boundary have been fully classified in $\mathbb{R}^3_+$ (Cavallotto, 2018). The main functional is: $$J_\alpha(E) = \mathcal{H}^{d}(E\setminus\Gamma) + \alpha\,\mathcal{H}^{d}(E\cap\Gamma),\quad \cos\theta_\alpha = \alpha,$$ where $\Gamma$ is the hyperplane $\{x_n=0\}$. All $2$D sliding minimal cones in $\mathbb{R}^3_+$ are either products of $1$D minimal cones or belong to four non-flat, one-parameter families: the $\mathbf{Y}_\beta$, $\overline{\mathbf{Y}}_\beta$, $\mathbf{W}_\beta$, and $\mathbf{T}_+$. The proof involves paired calibrations, tangent-cone tests, and explicit geometric construction. This completely resolves the structure of sliding minimal boundary cones in three dimensions and provides building blocks for higher-dimensional regularity (Cavallotto, 2018). In non-circular cones, all capillary minimal surfaces with $H\leq 0$ are radial graphs, but flatness need not always follow unless further symmetry or dimension restrictions hold (López et al., 2014).

5. Fractional Capillarity: Nonlocal Minimal Cones

In the fractional setting, the capillarity problem involves minimization of a nonlocal energy functional as developed in (Dipierro et al., 2020). For $n=2$, the only fractional minimal cones in the half-plane are angular sectors (wedges) with opening angle determined by the fractional Young law, which connects the wedge angle to the adhesion coefficient $\sigma$ and the fractional parameter $s$: $$1+\sigma = \frac{(\sin\vartheta)^s M(\vartheta,s)}{M(\pi/2,s)}.$$ No other non-flat sectors or minimal cones exist in the plane, as the translation method and boundary monotonicity exclude more complex configurations. In higher dimensions, the classification of nonlocal minimal cones (and hence fractional capillary cones) is still largely open with few known explicit examples.

6. Proof Techniques and Stability Analysis

Central analytic methods revolve around:

• Stability inequalities and Simons-type inequalities: For smooth capillary cones, a key stability criterion of Jerison–Savin type holds. For an admissible function $w$,

$$\int_M \left(|\nabla^M\varphi|^2 - |A|^2\varphi^2\right) \geq \int_\Gamma \cos\theta H_\Gamma \varphi^2,$$

where the right side captures the capillary boundary contribution (Pacati et al., 11 Feb 2025).

• Paired calibrations: For polyhedral or piecewise-linear cones (such as in sliding minimal cones), existence and minimality are proven via Lawlor–Morgan calibration strategies, ensuring the target cone achieves the variational lower bound and no local competitors can reduce energy (Cavallotto, 2018).
• ALE inversion and monotonicity: Radial graphical theorems use spherical inversion (an extension of the Alexandrov reflection method) and comparison of mean curvature post-inversion to conclude that minimal capillary surfaces with $H\leq 0$ in cones are necessarily radial graphs, and thus flat under further constraints (López et al., 2014).

7. Open Problems and Future Directions

The regularity threshold for non-flat area-minimizing capillary cones is now known to be $n\geq 7$ in general, analogous to the classical results for interior minimal surface singularities (Firester et al., 26 Jan 2026, Pacati et al., 11 Feb 2025). Major open directions include:

• Boundary singularities for general contact angles and dimension: Removing the non-sign-changing mean curvature or axial symmetry assumptions in $n \leq 6$ remains challenging (Pacati et al., 11 Feb 2025).
• Complete classification in higher codimension and for nonlocal functionals: Beyond explicit families with high symmetry, little is known about the full moduli space of singular area-minimizing capillary cones in large dimensions, or about fractional minimal cones for $n\geq 3$ (Dipierro et al., 2020).
• Connections to the one-phase problem: The limiting behavior of capillary cones as the contact angle $\theta \to 0$ connects to singular minimizers for the Alt–Caffarelli (Bernoulli) free boundary functional, elucidating singular boundary phenomena for classical and nonlocal free boundary problems (Firester et al., 26 Jan 2026).

Non-flat minimal capillary cones thus organize the boundary singularity theory for capillarity, offering both canonical models for singular points and crucial test cases for sharp regularity thresholds.