Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

TL;DR

This work introduces a nonlocal, capillary-boundary flow in the half-space with speed $\partial_t X=(\phi(t)-H^\alpha)\widetilde{\nu}$, where $\phi(t)$ is chosen to preserve either the enclosed volume or the capillary area. The authors develop a rigorous geometric-analytic framework, deriving evolution equations for $g_{ij}$, $\nu$, $h_{ij}$, and $H$, along with the capillary quantities $u$ and $\bar{u}$ under the operator $\mathcal{L}$, and show how boundary conditions interact with the nonlocal term. They establish a capillary pinching estimate, prove monotonicity of the capillary isoperimetric ratio, and obtain two-sided curvature bounds that yield global existence and smooth convergence to a spherical cap with the same preserved quantity. The results extend nonlocal, volume- or area-preserving curvature flows to capillary settings in the half-space and provide tools such as capillary Minkowski-type inequalities to study further capillarity problems. Overall, the paper demonstrates that convex capillary hypersurfaces under this nonlocal flow evolve for all time and converge to spheroidal caps, highlighting the role of capillary geometry in guiding long-time behavior.

Abstract

In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate that for any convex initial hypersurface with a capillary boundary, the flow exists for all time and smoothly converges to a spherical cap as $t \to +\infty$

Figures (1)

• Figure 1: Figure 1.

Theorems & Definitions (31)

• Theorem 1.1
• Proposition 2.1: WeX21
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 3.1
• proof