A constrained mean curvature flow and Alexandrov-Fenchel inequalities

1. Problem Statement

The paper addresses the geometric analysis of capillary hypersurfaces in the half-space $\bar{\mathbb{R}}^{n+1}_+$. Specifically, it investigates:

• The Isoperimetric Problem: Establishing inequalities relating the surface area and volume for hypersurfaces with a boundary constrained to the hyperplane $\partial \bar{\mathbb{R}}^{n+1}_+$ at a constant contact angle $\theta$.
• Alexandrov-Fenchel Inequalities: Proving a class of inequalities involving capillary quermassintegrals (generalized volumes) for convex capillary hypersurfaces.
• The Gap: While classical results exist for closed hypersurfaces (e.g., Huisken, Guan-Li) and some free boundary cases, the specific case of capillary boundaries (where the contact angle $\theta \in (0, \pi)$ is fixed but non-zero) required a tailored geometric flow to prove convergence and derive new inequalities.

2. Methodology

The authors employ a locally constrained mean curvature flow tailored specifically for capillary hypersurfaces.

A. The Geometric Flow

They introduce a new evolution equation for a family of hypersurfaces $x: M \times [0, T) \to \bar{\mathbb{R}}^{n+1}_+$:
$$\begin{cases} (\partial_t x)^\perp = \left( n + n \cos \theta \langle \nu, e \rangle - H \langle x, \nu \rangle \right) \nu, & \text{in } M \times [0, T), \\ \langle \nu, e \rangle = -\cos \theta, & \text{on } \partial M \times [0, T), \\ x(\cdot, 0) = x_0(\cdot), \end{cases}$$
where:

• $\nu$ is the unit normal, $H$ is the mean curvature.
• $e = -E_{n+1}$ is the outward normal of the boundary hyperplane.
• $\theta$ is the constant contact angle.
• The speed function $f = n(1 + \cos \theta \langle \nu, e \rangle) - H \langle x, \nu \rangle$ is derived from the Minkowski formula for capillary hypersurfaces.

B. Key Analytical Tools

1. Capillary Support Function: The authors define a modified support function $\bar{u} = \frac{\langle x, \nu \rangle}{1 + \cos \theta \langle \nu, e \rangle}$ to handle the boundary conditions effectively.
2. Evolution Equations: They derive parabolic evolution equations for geometric quantities (support function, mean curvature, harmonic curvature) under the flow.
3. Maximum Principle: Used extensively to establish:
   • Preservation of star-shapedness (via gradient estimates).
   • Preservation of convexity (specifically for $\theta \in (0, \pi/2]$).
   • Uniform $C^\infty$ estimates to ensure long-time existence.
4. Monotonicity: They utilize higher-order Minkowski formulas to show that specific geometric functionals (capillary quermassintegrals) evolve monotonically.

3. Key Contributions and Results

A. Long-Time Existence and Convergence (Theorem 1.2)

• Result: If the initial hypersurface is star-shaped with respect to the origin and the contact angle $\theta \in (0, \pi)$, the flow exists for all time $t \in [0, \infty)$.
• Convergence: The flow smoothly converges to a unique spherical cap $C_{\theta, r_0}$ (a portion of a sphere intersecting the boundary at angle $\theta$).
• Significance: Unlike previous works that were restricted to specific ranges of $\theta$, this result holds for the entire range $\theta \in (0, \pi)$.

B. Capillary Isoperimetric Inequality (Corollary 1.3)

Using the flow's convergence and the monotonicity of the capillary area functional $V_{1, \theta}$, they prove:
$$\frac{|\Sigma| - \cos \theta |\partial \Sigma|}{|S^n_\theta| - \cos \theta |\partial S^n_\theta|} \ge \left( \frac{|\bar{\Sigma}|}{|B^{n+1}_\theta|} \right)^{\frac{n}{n+1}}$$
where $|\Sigma|$ is the area, $|\partial \Sigma|$ is the boundary length, and $|\bar{\Sigma}|$ is the enclosed volume. Equality holds if and only if $\Sigma$ is a spherical cap.

C. Alexandrov-Fenchel Inequalities (Theorem 1.4)

For convex capillary hypersurfaces with $\theta \in (0, \pi/2]$, the authors establish a new class of inequalities for the capillary quermassintegrals $V_{k, \theta}$:
$$\left( \frac{V_{k, \theta}(\bar{\Sigma})}{b_\theta} \right)^{\frac{1}{n+1-k}} \ge \left( \frac{V_{0, \theta}(\bar{\Sigma})}{b_\theta} \right)^{\frac{1}{n+1}}, \quad \forall 1 \le k < n$$
where $b_\theta$ is the volume of the reference spherical cap.

• Significance: This provides a partial affirmative answer to a conjecture in [29] and generalizes classical Alexandrov-Fenchel inequalities to the capillary setting.

D. Volumetric Minkowski Inequality (Corollary 1.5)

As a direct consequence (setting $k=2$), they derive a volumetric Minkowski inequality for convex capillary hypersurfaces:
$$\int_\Sigma H dA - \sin \theta \cos \theta |\partial \Sigma| \ge n(n+1) b_\theta^{\frac{2}{n+1}} |\bar{\Sigma}|^{\frac{n-1}{n+1}}$$

4. Technical Highlights & Innovations

• Handling the Boundary: The introduction of the capillary support function $\bar{u}$ allows the authors to derive Neumann-type boundary conditions ($\nabla_\mu \bar{u} = 0$) which are crucial for applying the maximum principle near the boundary.
• Convexity Preservation: Proving that convexity is preserved requires $\theta \in (0, \pi/2]$. The authors analyze the evolution of the harmonic curvature $\bar{H}$ and show that the boundary term $\nabla_\mu \bar{H} \le 0$ holds under this angle restriction, preventing the loss of convexity.
• Monotonicity of Quermassintegrals: The flow is specifically designed so that the speed function $f$ makes the time derivative of the quermassintegrals non-positive (or zero for volume), directly linking the flow dynamics to the inequalities.

5. Significance

This paper bridges the gap between classical geometric flows for closed hypersurfaces and the more complex setting of capillary boundaries.

1. New Inequalities: It establishes a rigorous foundation for Alexandrov-Fenchel inequalities in the presence of capillary boundaries, a topic of growing interest in geometric analysis and mathematical physics.
2. Flow Proof: It provides a "flow proof" for these inequalities, offering a dynamic perspective that complements static variational methods.
3. Generality: The result covers the full range of contact angles for star-shaped surfaces, extending the scope of known results which were often limited to small angles or free boundaries.

In summary, the paper successfully utilizes a novel constrained mean curvature flow to solve the isoperimetric problem and prove deep geometric inequalities for capillary hypersurfaces, demonstrating the power of geometric flows in constrained boundary value problems.