A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

TL;DR

This paper introduces a new inverse curvature flow for convex hypersurfaces with capillary boundary in the half-space, proving convergence to spherical caps and establishing a comprehensive set of Alexandrov-Fenchel inequalities for such hypersurfaces.

Contribution

It develops a novel curvature flow approach and proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces with contact angle in (0, π/2].

Findings

Flow preserves convexity and exists for all time.

Flow converges smoothly to a spherical cap.

Establishes a complete family of Alexandrov-Fenchel inequalities.

Abstract

In this paper, we study the locally constrained inverse curvature flow for hypersurfaces in the half-space with $\theta$-capillary boundary, which was recently introduced by Wang-Weng-Xia. Assume that the initial hypersurface is strictly convex with the contact angle $\theta\in (0,\pi/2]$. We prove that the solution of the flow remains to be strictly convex for $t>0$, exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space with the contact angle $\theta\in(0,\pi/2]$. Along the proof, we develop a new tensor maximum principle for parabolic equations on compact manifold with proper Neumann boundary condition.