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A fully nonlinear locally constrained curvature flow for capillary hypersurface

Xinqun Mei, Liangjun Weng

TL;DR

The paper introduces a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in the half-space, aiming to unify and extend Alexandrov-Fenchel-type results via an evolution approach. The flow uses the curvature function $F=H_k^{1/k}$ with capillary boundary conditions, preserves convexity, and exists for all time, with smooth convergence to a spherical cap; a high-order capillary isoperimetric ratio is shown to be monotone along the flow. Central to the analysis are derived evolution equations, reduction to a scalar oblique PDE on star-shaped hypersurfaces, and a suite of $C^0$, $C^1$, and curvature estimates that yield global existence and convergence. As a byproduct, the monotonicity of quermassintegrals along the flow yields Alexandrov-Fenchel inequalities for capillary hypersurfaces in the half-space, with equality characterized by spherical caps.

Abstract

In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in \cite{MWW}. As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov-Fenchel inequalities.

A fully nonlinear locally constrained curvature flow for capillary hypersurface

TL;DR

The paper introduces a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in the half-space, aiming to unify and extend Alexandrov-Fenchel-type results via an evolution approach. The flow uses the curvature function with capillary boundary conditions, preserves convexity, and exists for all time, with smooth convergence to a spherical cap; a high-order capillary isoperimetric ratio is shown to be monotone along the flow. Central to the analysis are derived evolution equations, reduction to a scalar oblique PDE on star-shaped hypersurfaces, and a suite of , , and curvature estimates that yield global existence and convergence. As a byproduct, the monotonicity of quermassintegrals along the flow yields Alexandrov-Fenchel inequalities for capillary hypersurfaces in the half-space, with equality characterized by spherical caps.

Abstract

In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in \cite{MWW}. As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov-Fenchel inequalities.
Paper Structure (10 sections, 18 theorems, 94 equations)

This paper contains 10 sections, 18 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary symmetric functions
  4. Capillary hypersurface
  5. Evolution equations
  6. Long-time existence and convergence
  7. A scalar flow
  8. $C^{0}$ and $C^{1}$ estimates
  9. Curvature estimates
  10. Long-time existence and convergence

Key Result

Theorem 1.1

Let $\Sigma_0$ be a strictly convex capillary hypersurface in $\overline{\mathbb{R}^{n+1}_+}$ and $\theta\in(0,\frac{\pi}{2}]$, then the solution of flow flow with capillary starting with $\Sigma_0$ exists for all time. Moreover, $x(\cdot, t)$ smoothly converges to a uniquely determined spherical ca

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Ben
  • Lemma 2.5: Ben
  • Proposition 2.6: WeX21
  • Proposition 2.7
  • proof
  • ...and 20 more