Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

1. Problem Statement

The paper addresses the generalization of the Alexandrov-Fenchel inequalities and the Gauss-Bonnet-Chern formula to the setting of convex hypersurfaces with capillary boundary inside the unit Euclidean ball $B^{n+1}$.

• Context: Classical Alexandrov-Fenchel inequalities relate quermassintegrals (intrinsic volumes) of closed convex hypersurfaces in $\mathbb{R}^{n+1}$. Previous work (specifically Scheuer-Wang-Xia [42]) extended these to hypersurfaces with free boundary (intersecting the sphere $\partial B^{n+1}$ orthogonally).
• The Gap: This paper considers the more general $\theta$-capillary boundary condition, where the hypersurface $\Sigma$ intersects the boundary sphere $\partial B^{n+1}$ at a constant angle $\theta \in (0, \pi/2]$. This models the equilibrium shape of a liquid drop in a container with a specific contact angle.
• Objective:
  1. Define appropriate quermassintegrals ($W_{k,\theta}$) for hypersurfaces with $\theta$-capillary boundary.
  2. Derive their first variational formulas.
  3. Prove the Alexandrov-Fenchel type inequalities relating these functionals.
  4. Establish the Gauss-Bonnet-Chern formula for the highest-order quermassintegral.

2. Methodology

The authors employ a combination of geometric analysis, variational calculus, and geometric flows.

A. Definition of Geometric Functionals

The authors define a family of geometric functionals $W_{k,\theta}(\hat{\Sigma})$ for the domain $\hat{\Sigma}$ enclosed by the hypersurface $\Sigma$ and the boundary cap $\hat{\partial\Sigma}$ on the sphere.

• $W_{0,\theta}$ corresponds to the volume.
• $W_{1,\theta}$ corresponds to the energy functional involving area and wetting energy ($|\partial \Omega \cap B| - \cos\theta |\partial B \cap \Omega|$).
• Higher-order functionals $W_{k,\theta}$ involve integrals of normalized mean curvatures $H_k$ over $\Sigma$ and mixed terms involving the boundary $\partial\Sigma$ and the enclosed spherical cap $\hat{\partial\Sigma}$.
• Key Innovation: The definitions are constructed specifically so that their first variation formulas take a clean form involving the mean curvatures.

B. Minkowski-Type Formulas

A crucial step is the derivation of higher-order Minkowski formulas for capillary hypersurfaces.

• The authors utilize a conformal Killing vector field $X_e = \langle x, e \rangle x - \frac{1}{2}(|x|^2+1)e$ associated with a constant unit vector $e$.
• They prove the identity:
  $$\int_{\Sigma} H_{k-1} (\langle x, e \rangle + \cos\theta \langle \nu, e \rangle) dA = \int_{\Sigma} H_k \langle X_e, \nu \rangle dA$$
  This generalizes previous results for free boundaries to the capillary case.

C. Locally Constrained Curvature Flow

The core of the proof relies on a locally constrained inverse-type curvature flow:
$$\partial_t x = f \nu + T$$
where the normal speed is:
$$f = \frac{\langle x + \cos\theta \nu, e \rangle}{H_n/H_{n-1}} - \langle X_e, \nu \rangle$$

• Preservation Property: Using the Minkowski formulas and the first variational formulas, the authors show that under this flow:
  • $W_{n,\theta}$ is preserved (constant in time).
  • $W_{k,\theta}$ for $1 \le k \le n-1$ are non-decreasing.
• Convergence: The flow is shown to exist for all time and converge smoothly to a spherical cap (or flat disk) with the same contact angle $\theta$.
  • Barriers: The authors construct spherical caps as barriers to ensure the flow remains within the ball and maintains strict convexity.
  • Estimates: They derive uniform $C^\infty$ estimates (curvature bounds) by analyzing evolution equations for geometric quantities (like $F = H_n/H_{n-1}$ and mean curvature $H$) and utilizing the maximum principle.
  • Uniqueness: The limit is shown to be a unique spherical cap determined by the initial $W_{n,\theta}$ value.

3. Key Contributions and Results

Theorem 1.2 (Variational Formulas)

The paper establishes the first variational formulas for the defined quermassintegrals. For a normal variation with speed $f$:
$$\frac{d}{dt} W_{k,\theta}(\hat{\Sigma}_t) = \frac{n+1-k}{n+1} \int_{\Sigma_t} H_k f dA$$
and specifically, $\frac{d}{dt} W_{n+1,\theta} = 0$. This validates the definitions as the correct generalizations of quermassintegrals.

Theorem 1.3 (Alexandrov-Fenchel Inequalities)

For a convex hypersurface $\Sigma$ with $\theta$-capillary boundary:

1. Gauss-Bonnet-Chern Formula:
   $$W_{n+1,\theta}(\hat{\Sigma}) = \frac{\omega_n}{2(n+1)} I_{\sin^2\theta}\left(\frac{n}{2}, \frac{1}{2}\right)$$
   where $I$ is the regularized incomplete beta function. This value is constant for all such hypersurfaces (independent of shape, depending only on $\theta$).
2. Inequality: For $0 \le k \le n-1$:
   $$W_{n,\theta}(\hat{\Sigma}) \ge (f_n \circ f_k^{-1})(W_{k,\theta}(\hat{\Sigma}))$$
   where $f_k(r)$ is the value of $W_{k,\theta}$ for a spherical cap of radius $r$.
   • Equality Condition: Equality holds if and only if $\Sigma$ is a spherical cap or a flat disk with $\theta$-capillary boundary.

Corollary 1.4 (Case $n=2$)

The paper derives a specific Minkowski-type inequality for convex surfaces in $B^3$, providing an explicit formula relating the integral of $H_2$, boundary lengths, and the contact angle.

4. Significance

• Generalization: This work significantly extends the theory of geometric inequalities from free boundary ($\theta = \pi/2$) to the general capillary case ($\theta \in (0, \pi/2]$), which is physically more relevant for liquid drop problems.
• New Tools: The derivation of higher-order Minkowski formulas for capillary boundaries and the construction of the specific locally constrained flow provide new tools for studying geometric inequalities in domains with boundaries.
• Unified Framework: It unifies the study of isoperimetric problems, quermassintegrals, and curvature flows in the context of capillary surfaces, showing that spherical caps are the unique optimizers for these higher-order functionals.
• Methodological Advancement: The successful application of the locally constrained flow to prove these inequalities in the presence of a boundary condition (requiring careful handling of boundary terms and oblique derivative conditions) demonstrates the robustness of the curvature flow method in differential geometry.

In summary, Weng and Xia successfully define the correct geometric invariants for capillary hypersurfaces, prove they satisfy the Alexandrov-Fenchel inequalities, and identify spherical caps as the unique extremizers, thereby completing a major step in the geometric analysis of capillary surfaces in Euclidean balls.