Capillary Minimal Surfaces

• Capillary minimal surfaces are immersed surfaces with zero mean curvature in their interior and a fixed contact angle along the boundary.
• Existence proofs employ both direct minimization and min-max techniques, leading to classifications in convex domains and symmetric settings.
• Stability analyses include Robin-type boundary conditions and variational estimates that ensure robust regularity and sharp rigidity results.

A capillary minimal surface is a critical point of the area functional in a Riemannian manifold with boundary, with the additional feature that the surface meets the constraint boundary at a fixed contact angle. These surfaces occupy a central role in geometric analysis, connecting the theory of minimal surfaces, variational methods, and capillarity phenomena in physical and geometric contexts.

1. Definitions and Foundational Results

A connected, immersed surface $\Sigma$ in a smooth domain $U \subset \mathbb{R}^3$ (or more general Riemannian $3$-manifolds) with boundary $\partial\Sigma \subset \partial U$ is a capillary minimal surface at angle $\theta$ if:

• The mean curvature $H$ of $\Sigma$ vanishes identically in its interior: $H \equiv 0$.
• Along the boundary, $\Sigma$ meets $\partial U$ at a constant angle $\theta$, encoded by $n_U \cdot \nu = \cos\theta$, where $n_U$ is the outward unit normal of $\partial U$, and $\nu$ is the outward unit conormal of $\Sigma$ along $\partial\Sigma$. When $\theta = 90^\circ$, this is the free boundary condition $n_U \cdot \nu = 0$ (Yeon, 2019).

In variational terms, the capillarity functional for a smooth domain $M$ and region $\Omega \subset M$ is

$$A(\Omega) = \mathcal{H}^2(\partial\Omega \cap \text{int} M) + \cos\theta\,\mathcal{H}^2(\partial\Omega \cap \partial M),$$

with critical points corresponding to capillary minimal surfaces of angle $\theta$ (Li et al., 2021, Masi et al., 2021).

The first variation of this functional under tangential variations yields:

• Minimality: $H = 0$.
• Capillary boundary condition: $g(\nu, N_{\partial M}) = \cos\theta$, where $\nu$ is the normal to $\Sigma$, $N_{\partial M}$ is the normal to $\partial M$ (Li et al., 2021).

2. Existence, Classification, and Rigidity

Existence of capillary minimal surfaces under minimal (mean curvature zero) and fixed contact angle constraints is established via both direct minimization and min-max methods.

Existence and Min-max Theory:

A min-max approach, paralleling the Almgren–Pitts framework, constructs nontrivial, smooth, almost properly embedded minimal surfaces with arbitrary contact angle $\theta$ in compact Riemannian $3$-manifolds with boundary (Li et al., 2021, Masi et al., 2021). The min-max critical set is shown to consist of a (possibly disconnected) stationary varifold which, under appropriate regularity conditions (e.g., boundary regularity via almost minimization in annuli), yields a smooth capillary minimal surface up to the boundary.

Rigidity in the Ball Exterior:

A complete classification holds for embedded, finite total curvature capillary minimal surfaces in the exterior of the unit ball in $\mathbb{R}^3$ meeting $S^2$ orthogonally: the only possibilities are (up to rigid motion) (i) a flat disk in a plane, or (ii) a critical catenoid meeting $S^2$ perpendicularly. No Enneper-type ends occur if the flux on first homology vanishes. For exteriors of several disjoint spheres, again only planar solutions exist (Yeon, 2019).

Uniqueness in Convex Domains:

For compact, stable capillary minimal surfaces in the unit ball or more general convex domains, only planar disks or spherical caps are possible under the right-angle condition, by Hopf-type index arguments, the Jorge–Meeks theorem, and boundary maximum principles (Yeon, 2019, Naff et al., 14 Dec 2025).

Anisotropic Case:

Classification extends to capillary minimal surfaces with respect to anisotropic surface energies: in a half-space, the only weakly stable compact minimizers are truncated Wulff shapes, with the Bernstein-type rigidity (uniqueness of the flat half-plane) holding for properly immersed, strongly stable anisotropic capillary minimal surfaces with Euclidean area growth (Guo et al., 2023).

3. Stability, Morse Index, and Topology

The stability and index theory of capillary minimal surfaces generalizes the classical theory for closed and free-boundary minimal surfaces.

• The second variation formula incorporates interior and boundary contributions, with Robin-type boundary conditions reflecting the capillarity term:

$$Q(f, f) = \int_\Sigma (|\nabla f|^2 - (\text{Ric}(N) + |A|^2)f^2) \, dA - \int_{\partial\Sigma} q f^2 \, dL,$$

with $$q = (1/\sin\theta) \mathrm{II}(\bar\nu, \bar\nu) + (\cot\theta)A(\nu,\nu)$$ (Longa, 2021).

• For stable (index zero) capillary minimal surfaces in $(M,g)$, a sharp estimate holds:

$$I_\theta(\Sigma) = \frac{1}{2} \inf_M R_M \cdot |\Sigma| + \frac{1}{\sin\theta} \inf_{\partial M} H_{\partial M} \cdot |\partial\Sigma| \le 2\pi\chi(\Sigma),$$

where equality characterizes infinitesimal rigidity—totally geodesic surfaces with constant boundary curvatures (Longa, 2021).

Topologically, for index 1 surfaces, genus and number of boundary components are sharply bounded. Capillary minimal surfaces with finite strong index in Riemannian $3$-manifolds are conformally equivalent to compact surfaces with finitely many punctures corresponding to ends (Hong et al., 2021).

4. Regularity and Monotonicity

Boundary and interior regularity for capillary minimal surfaces (in the varifold sense) is now as robust as in the free-boundary or closed settings:

• Classical Allard-type $\epsilon$-regularity extends provided a sharp density bound and closeness to a model capillary half-plane are met. If the density at a boundary point plus the negative part of $\cos\beta$ is strictly less than one, then the surface is locally a $C^{1,\alpha}$ properly embedded hypersurface meeting the barrier at the prescribed angle (Masi et al., 31 May 2024).
• The regular set of a capillary varifold has full $(n-1)$-measure along the contact boundary when the density threshold holds (Masi et al., 31 May 2024).
• Monotonicity formulas tailored for capillary minimal surfaces in half-spaces or balls, generalizing Simon's formula, yield control on area and Willmore energy, guaranteeing Li-Yau-type rigidity: equality is attained only by totally umbilic (spherical cap or flat disk) surfaces (Wang et al., 5 Sep 2024).

Regularity Condition	Local Model	Regularity Result
$\Theta_V(0,1)+(\cos\beta(0))_{-}<1$	Capillary half-plane	$C^{1,\alpha}$, up to boundary
Mean curvature bounded, density condition	—	Open, dense $C^{1,\alpha}$ boundary set

These regularity results are critical for extracting smooth minimal varieties from variational or min-max procedures.

5. Variational Principles and Computational Aspects

Capillary minimal surfaces arise as stationary points of area plus "wetting" terms, subject to volume constraints or prescribed contact angle.

• Classical and anisotropic capillarity functionals:

$$F_Q(u) = \text{Area}(u) + \int Q(u) \cdot (u_x \wedge u_y) \, dx\,dy,$$

with $Q$ vanishing at infinity and $K = \mathrm{div}\,Q$ prescribing a spatially varying mean curvature (Caldiroli, 2014, Caldiroli et al., 2016). Volume constraints lead to Lagrange multiplier problems, whose solutions satisfy

$\Delta u = (K(u)-\lambda))(u_x \wedge u_y).$

Existence and nonexistence for minimizers, and behavior with respect to splitting (bubbling), are controlled by analysis of the capillarity term's size and sign, and concentration-compactness (Caldiroli, 2014, Caldiroli et al., 2016).

• In radially symmetric settings, the governing ODE for a surface of revolution in the Young-Laplace formulation is efficiently solved via Chebyshev pseudo-spectral methods, providing adaptive, robust numerical solvers for capillary tubes, sessile drops, and related interfaces. These collocation techniques outperform shooting methods, especially for large gaps or near-vertical contact angles (Treinen, 2022).

6. Special Cases and Generalizations

The theory encompasses several notable classes:

• Minimal Lagrangian capillary surfaces with Legendrian boundary in $B^4 \subset \mathbb{C}^2$: Only the equatorial plane disk (free boundary) and explicit "Lagrangian catenoids" (annulus-type, constant angle) exist as minimal Lagrangian capillary surfaces (Luo et al., 2020).
• Low-genus capillary minimal surfaces in spherical caps of $\mathbb{S}^3$ are classified: Disks are totally geodesic; annuli are nowhere umbilic and admit duality constructions. The Hopf-differential method generalizes Nitsche's and Fraser–Schoen's rigidity (Naff et al., 14 Dec 2025).
• The planar Plateau problem, when recast via capillarity with a nonlocal geometric potential, yields minimizers with positive thickness, resolving the classical "collapsing phenomenon" and guaranteeing embedded, $W^{1+s,p}$-regular solutions (Idu, 16 Oct 2024).

7. Impact, Open Problems, and Further Developments

The recent advances provide an overarching variational and analytic framework for capillary minimal surfaces with arbitrary boundary angles, robust regularity up to the boundary, precise classification in classical geometries, and sharp rigidity in both isotropic and anisotropic settings. Monotonicity formulas and stability theory now yield optimal area and Willmore bounds that interpolate between closed, free-boundary, and classical capillarity regimes.

Future directions include extension to higher codimension, mean curvature constraints, higher-genus surfaces (via index theory and min-max constructions), and further integration with geometric measure theory. The connection between singularity formation, density gaps, and boundary regularity remains particularly rich for further investigation (Masi et al., 31 May 2024, Wang et al., 5 Sep 2024, Naff et al., 14 Dec 2025).